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Suppose we are given the sequence $\{a_n\}_{n=1}^\infty$ defined by the rule $\forall n\in\mathbb{N}, a_n=n\sin \frac{\pi n}{4}$.

We can show that this is equivalent to:

$\forall n\in\mathbb{N},a_n= \begin{cases} \frac{n}{\sqrt{2}} & n \equiv 1\pmod{8} \lor n \equiv 3\pmod{8} \\ n & n \equiv 2\pmod{8} \\ 0 & n \equiv 0\pmod{8}\lor n \equiv 4\pmod{8} \\ -\frac{n}{\sqrt{2}} & n \equiv 5\pmod{8} \lor n \equiv 7\pmod{8} \\ -n & n\equiv 6 \pmod{8} \end{cases} $

Now the problem is to prove\show that around each number we take $L\in\mathbb{R}$ we can create\build a neighborhood such that:

  1. If $L$ does not equal to any element in the sequence $\{a_n\}_{n=1}^\infty$, Then we can build a neighborhood of $L$ such that no other element from the sequence will reside in it.

  2. If $L$ equals some element in the sequence $\{a_n\}_{n=1}^\infty$, Then, If $L$ equals zero (as zero is an element in the sequence), Then we can build a neighborhood of $L$ that will contain an infinite number of elements from the sequence, And If $L$ does not equals zero, Then we can build a neighborhood of $L$ that will contain exactly one element from the sequence.

Translating the problem into logic/set-theory notation we get that we have to show:

  1. If $\forall n\in\mathbb{N}, L\neq a_n$, Then $\exists \epsilon\in (0,\infty), |\{n\in\mathbb{N}|a_n\in N_\epsilon(L)\}|=0$

  2. If $\exists m\in\mathbb{N}, L= a_m$, Then,

    If $L=0$, Then $\exists \epsilon\in (0,\infty), |\{n\in\mathbb{N}|a_n\in N_\epsilon(L)\}|=\aleph_0$,

    And If $L\neq 0$, Then $\exists \epsilon\in (0,\infty), |\{n\in\mathbb{N}|a_n\in N_\epsilon(L)\}|=1$

Note: The notation $N_\epsilon (L)$ denotes the neighborhood of $L$ with radius $\epsilon$, I.e. $N_\epsilon(L)=(L-\epsilon,L+\epsilon)$


My try for proving 1:

Since in this case it is given that $\forall n\in\mathbb{N}, L\neq a_n$ we can conclude that $\forall n\in\mathbb{N}, a_n-L\neq 0$ and thus $\forall n\in\mathbb{N}, | a_n-L|>0$, Now If we define the set $A=\{|a_n-L| | n\in\mathbb{N}\}$ we get that this set is a non-empty set of real numbers that is bounded below by $0$ and thus its infimum $\inf(A)=\inf\limits_{n\in\mathbb{N}} |a_n-L|$ exists and satisfies $\inf\limits_{n\in\mathbb{N}}|a_n-L|\geq 0$ (as $0$ is a lower bound of $A$ and thus must be less than or equal to its infimum), Now if we just show that $\inf\limits_{n\in\mathbb{N}}|a_n-L|>0$ we could choose $\epsilon=\inf\limits_{n\in\mathbb{N}}|a_n-L|\in (0,\infty)$ and we would get that $\forall n\in\mathbb{N}, a_n\notin N_\epsilon(L)$ because if we suppose by contradiction that $\exists n\in\mathbb{N}, a_n\in N_\epsilon(L)$, Then we would get by the fact that $|a_n-L|\in A$ and by definition of infimum that $|a_n-L|\geq \inf(A)=\inf\limits_{n\in\mathbb{N}}|a_n-L|=\epsilon$ which contradicts the fact that $a_n\in N_\epsilon(L)$ (which is equivalent to $|a_n-L|<\epsilon$), Thus it must be the case that $\forall n\in\mathbb{N},a_n\notin N_\epsilon(L)$ and we can conclude that $\{n\in\mathbb{N}|a_n\in N_\epsilon(L)\}=\emptyset$ and thus $|\{n\in\mathbb{N}|a_n\in N_\epsilon(L)\}|=0$ as was to be shown.


My try for proving 2:

Since in this case it is given that $\exists m\in\mathbb{N}, L = a_m$, And since $0$ is in the image of $\{a_n\}_{n=1}^\infty$ (i.e. $0$ is an element in the set $\{a_n|n\in\mathbb{N}\}$) we get that there are two possibilities: $L=0$ or $L\neq 0$

If $L=0$, Any $\epsilon$ we choose from the set $(0,\infty)$ will work, As we can define the set $T=\{n\in\mathbb{N}|n\equiv 0 \pmod{8}\}$ and it is clear that for this set we have $\forall n\in T, a_n\in N_\epsilon(L)$ [because $\forall n\in T, a_n=0$ and because $0\in N_\epsilon(0)=N_\epsilon(L)$], And therefore we get that $T\subseteq \{n\in\mathbb{N}|a_n\in N_\epsilon(L)\}\subseteq \mathbb{N}$, But since $|T| = |\mathbb{N}| =\aleph_0$ we get that it must be the case that $|\{n\in\mathbb{N}|a_n\in N_\epsilon(L)\}|=\aleph_0$ as was to be shown.

Now if $L\neq 0$, We can define the set $A=\{|a_n-L| | m\neq n\in\mathbb{N}\}$, And we get that this set is a non-empty set of real numbers that is bounded below by $0$ and thus its infimum $\inf(A)=\inf\limits_{m\neq n\in\mathbb{N}} |a_n-L|$ exists and satisfies $\inf\limits_{m\neq n\in\mathbb{N}}|a_n-L|\geq 0$ (as $0$ is a lower bound of $A$ and thus must be less than or equal to its infimum), Now if we just show that $\inf\limits_{m\neq n\in\mathbb{N}}|a_n-L|>0$ we could choose $\epsilon=\inf\limits_{m\neq n\in\mathbb{N}}|a_n-L|\in (0,\infty)$ and we would get that $\forall m\neq n\in\mathbb{N}, a_n\notin N_\epsilon(L)$ because if we suppose by contradiction that $\exists m\neq n\in\mathbb{N}, a_n\in N_\epsilon(L)$, Then we would get by the fact that $|a_n-L|\in A$ and by definition of infimum that $|a_n-L|\geq \inf(A)=\inf\limits_{m\neq n\in\mathbb{N}}|a_n-L|=\epsilon$ which contradicts the fact that $a_n\in N_\epsilon(L)$ (which is equivalent to $|a_n-L|<\epsilon$), Thus it must be the case that $\forall m\neq n\in\mathbb{N},a_n\notin N_\epsilon(L)$ and we can conclude that $\{n\in\mathbb{N}|a_n\in N_\epsilon(L)\}=\{m\}$ (because $L=a_m$) and thus $|\{n\in\mathbb{N}|a_n\in N_\epsilon(L)\}|=1$ as was to be shown.


Thanks for any hint\help on how to prove that those infimums are indeed positive....

$\endgroup$
  • $\begingroup$ You have the right formalisms set up. To show your infimums are positive, you should think about whether you really need to take the infimum over everything in the sequence. One observation that helps here is that the absolute values of the nonzero terms grow with n, and you don't really need to look at anything with absolute value greater than 2L (why?) $\endgroup$ – jschnei Jul 13 '18 at 20:22
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Hints:

If $L$ is not an element of the sequence then there exists unique $n \in \mathbb{Z}$ such that $n < L < n+1$ and unique $m \in \mathbb{Z}$ such that $\frac{m}{\sqrt2} < L < \frac{m+1}{\sqrt2}$.

Therefore let $\varepsilon = \min\left\{\left|L-(n+1 )\right|, \left|L-n\right|, \left|L-\frac{m+1}{\sqrt2}\right|, \left|L - \frac{m}{\sqrt2}\right|\right\}$ and consider the neighbourhood $(L-\varepsilon, L + \varepsilon)$.

If $L = 0$ then any neighbourhood of $0$ will do, because $0$ appears infinitely many times in the sequence.

If $L$ appears in the sequence but $L \ne 0$, then:

  • if $L \in \mathbb{Z}$ then there exists a unique $m \in \mathbb{Z}$ such that $\frac{m}{\sqrt2} < L < \frac{m+1}{\sqrt2}$. Let $\varepsilon = \min\left\{1, \left|L-\frac{m+1}{\sqrt2}\right|, \left|L - \frac{m}{\sqrt2}\right|\right\}$.

  • if $L = \frac{n}{\sqrt2}$ for some $n\in \mathbb{Z}$ then there exists a unique $m \in \mathbb{Z}$ such that $m < L < m+1$. Let $\varepsilon = \min\left\{\frac1{\sqrt2}, \left|L-(m+1)\right|, \left|L - m\right|\right\}$.

and consider the neighbourhood $(L-\varepsilon, L + \varepsilon)$.

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Proof of Case (1):

We will show that indeed $\inf\limits_{n\in\mathbb{N}}|a_n-L|>0$:

Because $\forall n\in\mathbb{N}, L\neq a_n$, We get that in particular $L\neq 0$ and thus we have two possibilities: $L>0$ or $L<0$

If $L>0$, Then we can define an increasing sequence of natural numbers $\{n_k\}_{k=1}^\infty$ as follows: $\forall k\in\mathbb{N}, n_k := 8k+1$, And thus we can consider the subsequence $\{a_{n_k}\}_{k=1}^\infty$ of the sequence $\{a_n\}_{n=1}^\infty$ that corrsepsonds to this index sequence $\{n_k\}_{k=1}^\infty$, Now since $\forall k\in\mathbb{N}, n_k\equiv 1\pmod{8}$, We can conclude by definition of the sequence $\{a_n\}_{n=1}^\infty$ that $\forall k\in\mathbb{N}, a_{n_k}=\frac{n_k}{\sqrt{2}}$.

Now since it is clear that $\lim\limits_{k\to \infty} (8k+1)=\infty$, And since $\frac{1}{\sqrt{2}}>0$ we can conclude that $\lim\limits_{k\to\infty}\frac{8k+1}{\sqrt{2}}=\infty$, And thus $\lim\limits_{k\to\infty}\frac{n_k}{\sqrt{2}}=\infty$ which implies that $\lim\limits_{k\to\infty}a_{n_k}=\infty$.

Now by definition of limit at $\infty$, We get that $\exists K\in\mathbb{N},\forall K<k\in\mathbb{N},L<a_{n_k}$, Now if we choose any $k_0\in\mathbb{N}$ that satisfies $K<k_0$ we get that for this particular $k_0$ we have $L<a_{n_{k_0}}$.

Now let’s define $N:={n_{k_0}}+8$ (We add this $8$ inorder to ensure that $L$ will be an element in the set $B$ that will be defined below very soon), It is clear that $N\in\mathbb{N}$.

We will show that for this particular $N$ we have $N\equiv 1\pmod{8}$, $9\leq N$ and $L<a_N=\frac{N}{\sqrt{2}}$:

Since $n_{k_0}\equiv 1\pmod{8}$ and since $8\equiv 0\pmod{8}$ we get that $n_{k_0}+8\equiv 1\pmod{8}$, And therefore $N\equiv 1\pmod{8}$ as was to be shown.

Now since $n_{k_0}\in\mathbb{N}$, We get that $1\leq n_{k_0}$, And thus $9\leq n_{k_0}+8$, Which implies that $9\leq N$ as was to be shown.

Now by definition of the sequence $\{a_n\}_{n=1}^\infty$ and by using the fact $N\equiv 1\pmod{8}$ that we’ve just shown, We get that $a_N=\frac{N}{\sqrt{2}}$, Now since $0<8$, We get $n_{k_0}<n_{k_0}+8$, But because $N=n_{k_0}+8$, We conclude $n_{k_0}<N$, Now by dividing this inequality by $\sqrt{2}>0$ we get $\frac{n_{k_0}}{\sqrt{2}}<\frac{N}{\sqrt{2}}$, But since $a_{n_{k_0}}=\frac{n_{k_0}}{\sqrt{2}}$ and since $a_N=\frac{N}{\sqrt{2}}$, We can conclude that $a_{n_{k_0}}<a_N$, Now since $L<a_{n_{k_0}}$ we can conclude by transitivity of the $<$ relation that $L<a_N$ as was to be shown.

Now let’s define a set $B:=\{|a_n-L||n\in\{1,...,N\}\}$, Since $B$ is a non-empty finite set of real numbers we get that its minimum $\min(B)=\min\limits_{n\in\{1,...,N\}}|a_n-L|$ exists and is an element in $B$, I.e. $\min(B)=\min\limits_{n\in\{1,...,N\}}|a_n-L|\in B$, Now since $\forall n\in\mathbb{N}, |a_n-L|>0$ we get that in particular $\forall n\in\{1,...,N\},|a_n-L|>0$, Which implies that $\forall b\in B, b>0$, And thus, In particular for $\min(B)=\min\limits_{n\in\{1,...,N\}}|a_n-L|\in B$ we have $\min(B)=\min\limits_{n\in\{1,...,N\}}|a_n-L|>0$.

Now we’ll show that $L\in B$:

Since $4\leq 9$ and since $9\leq N$, We get that $4\leq N$, And thus $4\in\{1,...,N\}$, And we can conclude that $|a_4-L|\in B$, But since $4\equiv 4\pmod{8}$, We get $a_4=0$, And thus $|a_4-L|=|0-L|=|-L|=|L|$, And we conclude that $|L|\in B$, But because $L>0$ we get that $|L|=L$, And so $L\in B$ as was to be shown.

Now we’ll prove that $\forall n\in\mathbb{N}, \min\limits_{n\in\{1,...,N\}}|a_n-L|\leq |a_n-L|$:

Let $n\in\mathbb{N}$, Then there are two cases: $n\leq N$ or $N<n$

If $n\leq N$, Then we get that $n\in\{1,...,N\}$, And thus $|a_n-L|\in B$, And we can conclude that $\min\limits_{n\in\{1,...,N\}}|a_n-L|=\min(B)\leq |a_n-L|$.

If $N<n$, Then there are five cases: $a_n=0$, $a_n=\frac{n}{\sqrt{2}}$, $a_n=n$, $a_n=-\frac{n}{\sqrt{2}}$ or $a_n=-n$

If $a_n=0$, Then we get that $|a_n-L|=|0-L|=|-L|=|L|=L$, And as we’ve already shown that $L\in B$, We can conclude that $|a_n-L|\in B$, And thus $\min\limits_{n\in\{1,...,N\}}|a_n-L|=\min(B)\leq |a_n-L|$.

If $a_n=\frac{n}{\sqrt{2}}$, Then because $N<n$, We get by dividing this inequality by $\sqrt{2}>0$, That $\frac{N}{\sqrt{2}}<\frac{n}{\sqrt{2}}$, But since $a_N=\frac{N}{\sqrt{2}}$ we conclude that $a_N<a_n$, And thus (#) $a_N-L<a_n-L$, But because $L<a_N$ we get that $0<a_N-L$, Which implies that $0<a_n-L$, And thus $|a_N-L|=a_N-L$ And $|a_n-L|=a_n-L$ and we can conclude by (#) that $|a_N-L|<|a_n-L|$, Now because $|a_N-L|\in B$, We get that $\min\limits_{n\in\{1,...,N\}}|a_n-L|=\min(B)\leq |a_N-L|$, And we conclude that $\min\limits_{n\in\{1,...,N\}}|a_n-L|<|a_n-L|$.

If $a_n=n$, Then because $1<\sqrt{2}$, We get $\frac{1}{\sqrt{2}}<1$, And by multiplying this inequality by $N>0$, We get $\frac{N}{\sqrt{2}}<N$, But since $N<n$, We get $\frac{N}{\sqrt{2}}<n$, But since $a_N=\frac{N}{\sqrt{2}}$ we conclude that $a_N<a_n$, And thus (#) $a_N-L<a_n-L$, But because $L<a_N$ we get that $0<a_N-L$, Which implies that $0<a_n-L$, And thus $|a_N-L|=a_N-L$ And $|a_n-L|=a_n-L$ and we can conclude by (#) that $|a_N-L|<|a_n-L|$, Now because $|a_N-L|\in B$, We get that $\min\limits_{n\in\{1,...,N\}}|a_n-L|=\min(B)\leq |a_N-L|$, And we conclude that $\min\limits_{n\in\{1,...,N\}}|a_n-L|<|a_n-L|$.

If $a_n=-\frac{n}{\sqrt{2}}$, Then we get $|a_n-L|=|-\frac{n}{\sqrt{2}}-L|=|\frac{n}{\sqrt{2}}+L|$, But since $n>0$, We get that $\frac{n}{\sqrt{2}}>0$, And since $L>0$, We can conclude that $\frac{n}{\sqrt{2}}+L>0$, And so $|\frac{n}{\sqrt{2}}+L|=\frac{n}{\sqrt{2}}+L$, And we can conclude that $|a_n-L|=\frac{n}{\sqrt{2}}+L$, But since $\frac{n}{\sqrt{2}}>0$, We get that $\frac{n}{\sqrt{2}}+L>L$, And we can conclude that $|a_n-L|>L$, But since $L\in B$, We get that $\min\limits_{n\in\{1,...,N\}}|a_n-L|=\min(B)\leq L$, And thus $\min\limits_{n\in\{1,...,N\}}|a_n-L|< |a_n-L|$.

If $a_n=-n$, Then we get $|a_n-L|=|-n-L|=|n+L|$, But since $n>0$ and $L>0$, We get that $n+L>0$, And so $|n+L|=n+L$, And we can conclude that $|a_n-L|=n+L$, But since $n>0$, We get that $n+L>L$, And we can conclude that $|a_n-L|>L$, But since $L\in B$, We get that $\min\limits_{n\in\{1,...,N\}}|a_n-L|=\min(B)\leq L$, And thus $\min\limits_{n\in\{1,...,N\}}|a_n-L|< |a_n-L|$.

Thus we’ve shown that in each of those five cases we have $\min\limits_{n\in\{1,...,N\}}|a_n-L|\leq |a_n-L|$ as was to be shown.

Now we’ll prove that $\inf\limits_{n\in\mathbb{N}} |a_n-L|=\min\limits_{n\in\{1,...,N\}}|a_n-L|$ :

Since $\min\limits_{n\in\{1,...,N\}}|a_n-L|\in B$ and since $B\subseteq A$, We get that $\min\limits_{n\in\{1,...,N\}}|a_n-L|\in A$, But because what we’ve just shown, I.e. that $\forall n\in\mathbb{N}, \min\limits_{n\in\{1,...,N\}}|a_n-L|\leq |a_n-L|$, We conclude that $ \min\limits_{n\in\{1,...,N\}}|a_n-L|$ is the minimum element of $A$, I.e. $A$ has a minimum and its minimum satisfies $\min\limits_{n\in\mathbb{N}}|a_n-L|=\min(A)=\min\limits_{n\in\{1,...,N\}}|a_n-L|$, But because we’ve already shown that $\min\limits_{n\in\{1,...,N\}}|a_n-L|>0$, We conclude that $\min\limits_{n\in\mathbb{N}}|a_n-L|>0$, But because $\min\limits_{n\in\mathbb{N}}|a_n-L|=\inf\limits_{n\in\mathbb{N}}|a_n-L|$, We conclude that $\inf\limits_{n\in\mathbb{N}}|a_n-L|>0$ as was to be shown.

The case for $L<0$ can be proved in a similar way.

Q.E.D.


Proof of Case (2):

(Note: we consider only the case of $L\neq 0$ as we’ve already dealt with the case $L=0$ in the text of the question itself)

We will show that indeed $\inf\limits_{m\neq n\in\mathbb{N}}|a_n-L|>0$:

First we’ll prove that $\forall m\neq n\in\mathbb{N}, L\neq a_n$:

Suppose by contradiction that $\exists m\neq n\in\mathbb{N}, L = a_n$, Since $L=a_m$, We get that $a_n=a_m$, And thus by definition of the sequence $\{a_n\}_{n=1}^\infty$, We get that (#) $n \sin\frac{\pi n}{4}=m\sin\frac{\pi m}{4}$, Now since in this case $L\neq 0$ and since $L=a_n$ , We get that $a_n\neq 0$, And thus $n\sin\frac{\pi n}{4}\neq 0$, And we conclude that it must be the case that $\sin\frac{\pi n}{4}\neq 0$, Also since $m\in\mathbb{N}$ we get that $m\neq 0$, And we can conclude that $m \sin \frac{\pi n}{4}\neq 0$, Now by dividing the equality (#) by this non-zero number we get $\frac{n \sin\frac{\pi n}{4}}{m \sin \frac{\pi n}{4}}=\frac{m\sin\frac{\pi m}{4}}{m \sin \frac{\pi n}{4}}$ Which is equivalent to (##) $\frac{n}{m}=\frac{\sin \frac{\pi m}{4}}{\sin \frac{\pi n}{4}}$, Now since $\forall q\in\mathbb{N}, \sin\frac{\pi q}{4}=0,\pm 1,\pm \frac{1}{\sqrt{2}}$, We have in particular that $\sin\frac{\pi m}{4}=0,\pm 1,\pm \frac{1}{\sqrt{2}}$ and $\sin\frac{\pi n}{4}=0,\pm 1,\pm \frac{1}{\sqrt{2}}$, But since we’ve already shown that $\sin\frac{\pi n}{4}\neq 0$, We get that $\sin\frac{\pi n}{4}=\pm 1,\pm \frac{1}{\sqrt{2}}$, Now we’ll show that $\sin\frac{\pi m}{4}\neq 0$, Because If we suppose by contradiction that indeed $\sin\frac{\pi m}{4}=0$ , Then we get by equation (##) that $\frac{n}{m}=\frac{0}{sin\frac{\pi n}{4}} = 0$ and thus $n=m\times 0=0$ which contradicts the fact that $n\in\mathbb{N}$, Thus it must be the case that $\sin\frac{\pi m}{4}\neq 0$, And we conclude that $\sin\frac{\pi m}{4}=\pm 1,\pm \frac{1}{\sqrt{2}}$, And therefore $\frac{\sin\frac{\pi m}{4}}{\sin\frac{\pi n}{4}}=\frac{\pm 1}{\pm 1},\frac{\pm 1}{\pm\frac{1}{\sqrt{2}}},\frac{\pm\frac{1}{\sqrt{2}}}{\pm 1},\frac{\pm\frac{1}{\sqrt{2}}}{\pm\frac{1}{\sqrt{2}}}=\pm 1, \pm \sqrt{2}, \pm \frac{1}{\sqrt{2}}$, And by equality (##) we conclude that $\frac{n}{m}=\pm 1,\pm \sqrt{2},\pm \frac{1}{\sqrt{2}}$, Now since $m,n\in\mathbb{N}$ we get that $\frac{n}{m}$ must be a rational number, I.e. $\frac{n}{m}\in\mathbb{Q}$, And thus it must be the case that $\frac{n}{m}\neq \pm \sqrt{2},\pm\frac{1}{\sqrt{2}}$, Also we’ll show that $\frac{n}{m}\neq -1$, Because if we suppose by contradiction that indeed $\frac{n}{m}=-1$, Then we would get that $n=-m$, But since $n\in\mathbb{N}$ we get that $0<n$ and thus $0<-m$ which implies that $m<0$, But since $m\in\mathbb{N}$ we get that $0<m$ and therefore we got thatA $0<m<0$ which implies that $0<0$ and we’ve reached a contradiction, Thus it must be the case that $\frac{n}{m}\neq -1$.

In summary we’ve shown that it must be the case that $\frac{n}{m}=\pm 1,\pm \sqrt{2},\pm \frac{1}{\sqrt{2}}$, And that it must be the case that $\frac{n}{m}\neq -1,\pm \sqrt{2},\pm \frac{1}{\sqrt{2}}$, And therefore we conclude that it must be the case that $\frac{n}{m}=1$ and we conclude that $m=n$, But this contradicts the fact that $m\neq n$, And thus it must be the case that $\forall m\neq n\in\mathbb{N}, L\neq a_n$ as was to be shown.

Now since $L\neq 0$ we get that there are two possibilities: $L>0$ or $L<0$

If $L>0$, Then we can define an increasing sequence of natural numbers $\{n_k\}_{k=1}^\infty$ as follows: $\forall k\in\mathbb{N}, n_k := 8k+1$, And thus we can consider the subsequence $\{a_{n_k}\}_{k=1}^\infty$ of the sequence $\{a_n\}_{n=1}^\infty$ that corrsepsonds to this index sequence $\{n_k\}_{k=1}^\infty$, Now since $\forall k\in\mathbb{N}, n_k\equiv 1\pmod{8}$, We can conclude by definition of the sequence $\{a_n\}_{n=1}^\infty$ that $\forall k\in\mathbb{N}, a_{n_k}=\frac{n_k}{\sqrt{2}}$.

Now since it is clear that $\lim\limits_{k\to \infty} (8k+1)=\infty$, And since $\frac{1}{\sqrt{2}}>0$ we can conclude that $\lim\limits_{k\to\infty}\frac{8k+1}{\sqrt{2}}=\infty$, And thus $\lim\limits_{k\to\infty}\frac{n_k}{\sqrt{2}}=\infty$ which implies that $\lim\limits_{k\to\infty}a_{n_k}=\infty$.

Now by definition of limit at $\infty$, We get that $\exists K\in\mathbb{N},\forall K<k\in\mathbb{N},L<a_{n_k}$, Now if we choose any $k_0\in\mathbb{N}$ that satisfies $K<k_0$ we get that for this particular $k_0$ we have $L<a_{n_{k_0}}$.

Now let’s define $M:=m+9-m\mod 8$, We’ll show that $m<M\in\mathbb{N}$ and that $M\mod 8=1$:

Since $0\leq m\mod 8\leq 7$, We get that $-7\leq -m\mod 8\leq 0$, And by adding $9$ we get $2=9-7\leq 9-m\mod 8\leq 9+0=9$, Now by adding $m$ we get $m+2\leq m+9-m\mod 8\leq m+9$, And thus $m+2\leq M\leq m+9$, But since $0<2$ we get that $m<m+2$, And we can conclude that $m<M$, Also since $m,9,m\mod 8\in\mathbb{Z}$ we conclude that $m+9-m\mod 8\in\mathbb{Z}$ and thus $M\in\mathbb{Z}$, Now since $m\in\mathbb{N}$ we get that $0<m$, And by the fact $m<M$ that we’ve just shown, We conclude that $0<M$, And thus $M\in\mathbb{N}$, And in summary we’ve got that $m<M\in\mathbb{N}$ as was to be shown.

Now we have:

(###) $M\mod 8 = (m+9-m\mod 8)\mod 8=((m+9)\mod 8- (m \mod 8)\mod 8)\mod 8=((m\mod 8+9\mod 8)\mod 8 - (m\mod 8))\mod 8=((m\mod 8+1)\mod 8-m\mod 8)\mod 8$,

And now there are two possibilities $0\leq m\mod 8\leq 6$ or $m\mod 8=7$:

If $0\leq m\mod 8\leq 6$, Then $1\leq m\mod 8+1\leq 7$, And thus $(m\mod 8+1)\mod 8=m\mod 8+1$, And we conclude by (###) that $M\mod 8=(m\mod 8+ 1-m\mod 8)\mod 8=1\mod 8=1$.

If $m\mod 8=7$, Then $M\mod 8= ((7+1)\mod 8-7)\mod 8=(8\mod 8-7)\mod 8=(0-7)\mod 8 =(-7)\mod 8=1$.

Thus we see that it is always the case that $M\mod 8=1$ as was to be shown.

Now let’s define $N:=\max (M,{n_{k_0}})+8$ (We add this $8$ inorder to ensure that $L$ will be an element in the set $B$ that will be defined below very soon), Since $M,n_{k_0}\in\mathbb{N}$, We get that $\max(M,n_{k_0})\in\mathbb{N}$, And since $8\in\mathbb{N}$, We can conclude that $\max(M,n_{k_0})+8\in\mathbb{N}$, And thus $N\in\mathbb{N}$.

We will show that for this particular $N$ we have $N\equiv 1\pmod{8}$, $9\leq N$, $L<a_N=\frac{N}{\sqrt{2}}$ and $m<N$:

Since $M\equiv 1\pmod{8}$, And Since $n_{k_0}\equiv 1\pmod{8}$, We get that $\max(M,n_{k_0})\equiv 1\pmod{8}$, Also since $8\equiv 0\pmod{8}$ we get that $\max(M,n_{k_0})+8\equiv 1\pmod{8}$, And therefore $N\equiv 1\pmod{8}$ as was to be shown.

Now since $\max(M,n_{k_0})\in\mathbb{N}$, We get that $1\leq \max (M,n_{k_0})$, And thus $9\leq \max (M,n_{k_0})+8$, Which implies that $9\leq N$ as was to be shown.

Now by definition of the sequence $\{a_n\}_{n=1}^\infty$ and by using the fact $N\equiv 1\pmod{8}$ that we’ve just shown, We get that $a_N=\frac{N}{\sqrt{2}}$, Now since $0<8$, We get $\max (M,n_{k_0})<\max (M,n_{k_0})+8$, But because $N=\max (M,n_{k_0})+8$, We conclude $\max(M,n_{k_0})<N$, But Since $n_{k_0}\leq \max (M,n_{k_0})$, We get that $n_{k_0}<N$, Now by dividing this inequality by $\sqrt{2}>0$ we get $\frac{n_{k_0}}{\sqrt{2}}<\frac{N}{\sqrt{2}}$, But since $a_{n_{k_0}}=\frac{n_{k_0}}{\sqrt{2}}$ and since $a_N=\frac{N}{\sqrt{2}}$, We can conclude that $a_{n_{k_0}}<a_N$, Now since $L<a_{n_{k_0}}$ we can conclude that $L<a_N$ as was to be shown.

Now since $0<8$, We get $\max (M,n_{k_0})<\max (M,n_{k_0})+8$, But because $N=\max (M,n_{k_0})+8$, We conclude $\max(M,n_{k_0})<N$, But since $M\leq \max (M,n_{k_0})$, We get that $M<N$, But because we’ve already shown that $m<M$ we can conclude that $m<N$ as was to be shown.

Now let’s define a set $B:=\{|a_n-L||n\in\{1,...,N\}\backslash\{m\}\}$, We’ll show that $L\in B$ and that $|a_N-L|\in B$:

Since $4\leq 9$ and since $9\leq N$, We get that $4\leq N$, And thus $4\in\{1,...,N\}$, Now we’ll show that $4\notin\{m\}$, Because if we suppose by contradiction that $4\in\{m\}$, Then we’ll get that $4=m$, And thus $a_m=a_4$, But since $4\equiv 4 \pmod{8}$, We get by definition of the sequence $\{a_n\}_{n=1}^\infty$ that $a_4=0$, And thus $a_m=0$, But since $a_m=L$ we get that $L=0$ which contradicts the fact that $L\neq 0$, And thus it must be the case that $4\notin\{m\}$, And we can conclude that $4\in\{1,...,N\}\backslash\{m\}$, And therefore we can conclude that $|a_4-L|\in B$, But since $|a_4-L|=|0-L|=|-L|=|L|$, We conclude that $|L|\in B$, But because $L>0$ we get that $|L|=L$, And so $L\in B$ as was to be shown

Also since $N\in\{1,...,N\}$ And since $N\notin\{m\}$, Because if we suppose by contradiction that $N\in\{m\}$, Then we’ll get that $N=m$, But as we’ve already shown that $m<N$, We conclude that $m<m$ and we’ve reached a contradiction, Thus it must be the case that $N\notin\{m\}$, And we can conclude that $N\in\{1,...,N\}\backslash \{m\}$, And thus $|a_N-L|\in B$ as was to be shown.

Now since $B$ is a non-empty (as we’ve just shown that $L\in B$) finite set of real numbers we get that its minimum $\min(B)=\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|$ exists and is an element in $B$, I.e. $\min(B)=\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|\in B$, Now since we’ve shown that $\forall m\neq n\in\mathbb{N}, a_n\neq L$, We conclude that $\forall m\neq n\in\mathbb{N}, a_n-L\neq 0$, Which implies that $\forall m\neq n\in\mathbb{N}, |a_n-L|>0$, And thus we get that in particular $\forall n\in\{1,...,N\}\backslash\{m\},|a_n-L|>0$, Which implies that $\forall b\in B, b>0$, And thus, In particular for $\min(B)=\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|\in B$ we have $\min(B)=\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|>0$.

Now we’ll prove that $\forall m\neq n\in\mathbb{N}, \min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|\leq |a_n-L|$:

Let $m\neq n\in\mathbb{N}$, Then there are two cases: $n\leq N$ or $N<n$

If $n\leq N$, Then we get that $n\in\{1,...,N\}$, But as $n\neq m$, We get that $n\notin \{m\}$, And thus $n\in\{1,...,N\}\backslash\{m\}$, And we conclude that $|a_n-L|\in B$, And therefore we can conclude that $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|=\min(B)\leq |a_n-L|$.

If $N<n$, Then there are five cases: $a_n=0$, $a_n=\frac{n}{\sqrt{2}}$, $a_n=n$, $a_n=-\frac{n}{\sqrt{2}}$ or $a_n=-n$

If $a_n=0$, Then we get that $|a_n-L|=|0-L|=|-L|=|L|=L$, And as we’ve already shown that $L\in B$, We can conclude that $|a_n-L|\in B$, And thus $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|=\min(B)\leq |a_n-L|$.

If $a_n=\frac{n}{\sqrt{2}}$, Then because $N<n$, We get by dividing this inequality by $\sqrt{2}>0$, That $\frac{N}{\sqrt{2}}<\frac{n}{\sqrt{2}}$, But since $a_N=\frac{N}{\sqrt{2}}$, We conclude that $a_N<a_n$, And thus (§) $a_N-L<a_n-L$, But because $L<a_N$, We get that $0<a_N-L$, Which implies that $0<a_n-L$, And thus $|a_N-L|=a_N-L$ and $|a_n-L|=a_n-L$, And we can conclude by (§) that $|a_N-L|<|a_n-L|$, Now because we’ve already shown that $|a_N-L|\in B$, We get that $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|=\min(B)\leq |a_N-L|$, And we conclude that $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|<|a_n-L|$.

If $a_n=n$, Then because $1<\sqrt{2}$, We get $\frac{1}{\sqrt{2}}<1$, And by multiplying this inequality by $N>0$, We get $\frac{N}{\sqrt{2}}<N$, But since $N<n$, We get $\frac{N}{\sqrt{2}}<n$, But since $a_N=\frac{N}{\sqrt{2}}$, We conclude that $a_N<a_n$, And thus (§) $a_N-L<a_n-L$, But because $L<a_N$, We get that $0<a_N-L$, Which implies that $0<a_n-L$, And thus $|a_N-L|=a_N-L$ and $|a_n-L|=a_n-L$, And we can conclude by (§) that $|a_N-L|<|a_n-L|$, Now because we’ve already shown that $|a_N-L|\in B$, We get that $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|=\min(B)\leq |a_N-L|$, And we conclude that $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|<|a_n-L|$.

If $a_n=-\frac{n}{\sqrt{2}}$, Then we get $|a_n-L|=|-\frac{n}{\sqrt{2}}-L|=|\frac{n}{\sqrt{2}}+L|$, But since $n>0$, We get that $\frac{n}{\sqrt{2}}>0$, And since $L>0$, We can conclude that $\frac{n}{\sqrt{2}}+L>0$, And so $|\frac{n}{\sqrt{2}}+L|=\frac{n}{\sqrt{2}}+L$, And we can conclude that $|a_n-L|=\frac{n}{\sqrt{2}}+L$, But since $\frac{n}{\sqrt{2}}>0$, We get that $\frac{n}{\sqrt{2}}+L>L$, And we can conclude that $|a_n-L|>L$, But since $L\in B$, We get that $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|=\min(B)\leq L$, And thus $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|< |a_n-L|$.

If $a_n=-n$, Then we get $|a_n-L|=|-n-L|=|n+L|$, But since $n>0$ and $L>0$, We get that $n+L>0$, And so $|n+L|=n+L$, And we can conclude that $|a_n-L|=n+L$, But since $n>0$, We get that $n+L>L$, And we can conclude that $|a_n-L|>L$, But since $L\in B$, We get that $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|=\min(B)\leq L$, And thus $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|< |a_n-L|$.

Thus we’ve shown that in each of those five cases we have $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|\leq |a_n-L|$ as was to be shown.

Now we’ll prove that $\inf\limits_{m\neq n\in\mathbb{N}} |a_n-L|=\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|$ :

Since $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|\in B$ and since $B\subseteq A$, We get that $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|\in A$, But because what we’ve just shown, I.e. that $\forall m\neq n\in\mathbb{N}, \min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|\leq |a_n-L|$, We conclude that $ \min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|$ is the minimum element of $A$, I.e. $A$ has a minimum and its minimum satisfies $\min\limits_{m\neq n\in\mathbb{N}}|a_n-L|=\min(A)=\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|$, But because we’ve already shown that $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|>0$, We conclude that $\min\limits_{m\neq n\in\mathbb{N}}|a_n-L|>0$, But because $\min\limits_{m\neq n\in\mathbb{N}}|a_n-L|=\inf\limits_{m\neq n\in\mathbb{N}}|a_n-L|$, We conclude that $\inf\limits_{m\neq n\in\mathbb{N}}|a_n-L|>0$ as was to be shown.

The case for $L<0$ can be proved in a similar way.

Q.E.D.


Practically Calculating the Infimums:

If we want to practically calculate the infimums $\inf\limits_{n\in\mathbb{N}}|a_n-L|$ and $\inf\limits_{m\neq n\in\mathbb{N}}|a_n-L|$ that we got in cases (1) and (2) (only the case in which $L\neq 0$) respectively, Then we will make use of the facts that in case (1) we have $\inf\limits_{n\in\mathbb{N}} |a_n-L|=\min\limits_{n\in\{1,...,N\}}|a_n-L|$ and in case (2) (only the case in which $L\neq 0$) we have $\inf\limits_{m\neq n\in\mathbb{N}} |a_n-L|=\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|$, Therefore inorder to get the value of the appropriate infimum we will just calculate the value of $\min\limits_{n\in\{1,...,N\}}|a_n-L|$ or $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|$ depending on wether we are dealing with case (1) or case (2) (only the case in which $L\neq 0$).

We will consider only the case of $L>0$:

First let’s define $max\_index := \lceil\sqrt{2}L\rceil$, We’ll show that $max\_index\in\mathbb{N}$:

Since $L>0$ and $\sqrt{2}>0$, We get that $\sqrt{2}L>0$, And thus $0<\lceil \sqrt{2}L\rceil\in\mathbb{Z}$, And therefore we can conclude that $\lceil \sqrt{2}L\rceil\in\mathbb{N}$, And thus $max\_index\in\mathbb{N}$ as was to be shown.

Now we’ll prove that (#) $\forall max\_index < n\in\mathbb{N}, L\neq a_n$:

Let $max\_index <n\in\mathbb{N}$, Then by definition of $max\_index$, We get that $\lceil\sqrt{2}L\rceil<n$, But since $\sqrt{2}L\leq\lceil\sqrt{2}L\rceil$, We conclude that $\sqrt{2}L<n$, And thus (§) $L<\frac{n}{\sqrt{2}}$.

Now there are five cases for $a_n$: $a_n=0$, $a_n=\frac{n}{\sqrt{2}}$, $a_n=n$, $a_n=-\frac{n}{\sqrt{2}}$ or $a_n=-n$

If $a_n=0$, Then we can conclude by the fact $L>0$ that $L>a_n$, And thus in particular $L\neq a_n$.

If $a_n=\frac{n}{\sqrt{2}}$, Then we can conclude by (§) that $L<\frac{n}{\sqrt{2}}$, And thus $L<a_n$, And we get that in particular $L\neq a_n$.

If $a_n=n$, Then we can conclude by (§) that $L<\frac{n}{\sqrt{2}}$, But since $1<\sqrt{2}$, We get $\frac{1}{\sqrt{2}}<1$, And by multiplying this inequality by $n>0$ (as $n\in\mathbb{N}$), We get that $\frac{n}{\sqrt{2}}<n$, And thus we can conclude that $L<n$, Which implies that $L<a_n$, And thus in particular $L\neq a_n$.

If $a_n=-\frac{n}{\sqrt{2}}$, Then we get by the fact $0<\frac{n}{\sqrt{2}}$ that $-\frac{n}{\sqrt{2}}<0$, And thus $a_n<0$, But since $0<L$, We conclude that $a_n<L$, And thus in particular we have $L\neq a_n$.

If $a_n=-n$, Then we get by the fact $0<n$ that $-n<0$, And thus $a_n<0$, But since $0<L$, We conclude that $a_n<L$, And thus in particular we have $L\neq a_n$.

Thus we’ve shown that in each of those five cases we have $L\neq a_n$ as was to be shown.

By (#) we conclude that in order for us to check wether $\exists m\in\mathbb{N}, L=a_m$, It is enough for us to check wether $\exists m\in\{1,...,max\_index\},L=a_m$.

By taking this observation into account, We can program an algorithm that will determine if $L$ equals an element in the sequence $\{a_n\}_{n=1}^\infty$, And if it is, It will return the index of the first occurrence of $L$ in the sequence $\{a_n\}_{n=1}^\infty$, And if not it will return an index of $-1$, I.e. a non-existing index that will indicate that $L$ does not equal any element of the sequence $\{a_n\}_{n=1}^\infty$.

// Checks wether the argument equals an element in the sequence,
// If it is, The function will return the index in which it was found,
// And if it was not, It will return -1.

int search_for(double L)
{
    // The maximal index that must be examined:
    int max_index = ceil(sqrt(2) * L);

    // The index at which it was found (if was found):
    int index = 1;

    // Indicates wether the value was found at the current interation:
    bool is_found = false;

    do
    {
        if(a(index) == L)
            is_found = true;
        else
            index++;
    }
    while (!is_found  && index <= max_index);

    if (!is_found)
        index = -1;

    return index;
}

Now, We will use this algorithm in order to check wether $L$ equals an element in the sequence, If it is not, Then we are dealing with case (1) and the following algorithm will return the value of $\min\limits_{n\in\{1,...,N\}}|a_n-L|$, And if it is, Then we are dealing with case (2) (with $L\neq 0$) And the following algorithm will return the value of $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|$, And thus, We will get the exact values of the infimums.

double infimum(double L)
{
   double minimal_distance; // Will hold in the end the value of the appropriate infimum.
   int m; // Holds the first index for which $L=a_m$ if any.
   int k0; // Holds the index that its existence was guaranteed in cases (1) and (2).
   int nk0; // Holds the element of the sequence $\{n_k\}$ at the index $k_0$. 
   int M; // Holds the value of $M$ as defined in case (2) [Used only if we are dealing with case (2)].
   int N; // Holds the greatest index that must be considered inorder to evaluate the minimums.

   // We have to choose an index $k_0$ that will ensure that $L<a_{n_{k_0}}$, 
   // And the $k_0$ defined this way will satisfy it.
   k0 = ceil((sqrt(2)*L-1)/8)+1;

   // The $k_0$’s element of the sequence $\{n_k\}$.
   nk0 = 8*k0+1;

   // Searches for the first index for which $L=a_m$ if any.
   m = search_for(L);

   // L equals some element in the sequence and thus $m>0$:
   if (m > 0)
   {
       // Since we are dealing here with case (2), 
       // We will define $M$ and $N$ as they were defined in case (2).
       M = m + 9 - m%8;
       N = max(M,nk0)+8;
   }

   // L does not equal any element in the sequence and thus $m=-1$.
   else
   {
       // Since we are dealing here with case (1),
       // We will define $N$ as it was defined in case (1).
       N = nk0 + 8;
   }

   // The following chunk of code will calculate the appropriate minimum based   
   // on wether we are dealing with case (1) or case (2).
   // Note:
   // If we are dealing with case (1), Then we know that $m=-1$, 
   // And thus the condition $i\neq m$ in the for loop header will never be satisfied,
   // And in the end $minimal_distance$ will hold the value $\min\limits_{n\in\{1,...,N\}}|a_n-L|$
   // And if we are dealing with case (2), Then we know that $m\in\{1,...,N\}$,
   // And thus the condition $i\neq m$ in the for loop header will skip over the
   // index $m$, And in the end $minimal_distance$ will hold the value  $\min\limits_{n\in\{1,...,N\}\backslash\{m\}}|a_n-L|$ 
   minimal_distance = INFINITY;
   for (int i = 1; i != m && i <= N; i++)
   {
       double current_distance = abs(a(i) - L);
       if (current_distance < minimal_distance)
           minimal_distance = current_distance;
   }

   return minimal_distance;
}

Note: The algorithms work only for the case $L>0$ and for the case $L<0$ some modifications are needed inorder for the algorithm to work in this case.

By putting the whole program together we get:

https://github.com/MathNerd/Projects/blob/master/calculate_infimums.c

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