For questions on suprema and infima. Use together with a subject area tag, such as (real-analysis) or (order-theory).

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Prove: Monotonic And Bounded Sequence- Converges

Let $a_n$ be a monotonic and bounded sequence, WLOG let assume it is monotonic increasing. $a_n$ is bounded therefore there is a Supremum, $Sup(a_n)=a$, therefore $a_n<a+\epsilon$. On the other ...
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0answers
18 views

Proof concerning nested intervals property

I have read and believe to understand the Nested Intervals property theorem but am not sure how to approach the following proof. (The theorem it refers to is just the Nested intervals theorem). ...
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1answer
81 views

The importance of being real

Let $\Sigma$ be a collection of holomorphic, one-to-one function from some simply connected region $\Omega$, which map $\Omega$ into the open unit disc $U$. Fix $z_0 \in \Omega$ and put $$\eta = ...
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1answer
24 views

If the set $B=\{f(x) : x\in A\}$ has supremum and $C=\{k+f(x): x\in A\}$, then what is $\sup C$?

If the set $B=\{f(x) : x\in A\}$ has supremum and $C=\{k+f(x): x\in A\}$, then what is $\sup C$? Since $C$ is not an empty set and $f(x)\le \sup(B) ⇒ k+f(x)\le k+\sup(B)$. So $C$ is bounded ...
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2answers
26 views

Monotonous everywhere function

$f: \mathbb R \to \mathbb R,\forall x \in \mathbb R $ $\exists \delta \gt 0 : f$ is non-decreasing on $(x-\delta,x+\delta)$(I call that statement A). I need to prove that $f$ is non-decreasing on ...
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2answers
43 views

Sup and inf of $n \sin(1/n)$

If $n$ is a natural number then, what is the supremum and infimum of $n\sin(1/n)$? is the question I want to solve. I drew $sin(x)/x$ graph and I think that the supremum is $1$ and infimum is ...
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2answers
54 views

Calculate $\sup\limits_{x\in(0,+\infty)}\frac{x^2 e^{-n/x}}{n^2+x^2}$

Calculate $$\sup_{x\in(0,+\infty)}\frac{x^2 e^{-n/x}}{n^2+x^2}$$ The derivative is $$\frac{n e^{-n/x} (n+x)^2}{(n^2+x^2)^2}\geq 0$$ then $$\sup_{x\in(0,+\infty)}\frac{x^2 ...
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2answers
34 views

Limit of monotonic function

I have to prove that if $(x_1 \gt x_2) \Rightarrow (f(x_1) \ge f(x_2))$, then $$\forall a \in \mathbb R \exists L \in \mathbb R \lim_{x \to a^+}f(x) = L$$ I have a feeling that L = $inf_{x \in (a,a+ ...
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0answers
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Let A= [0,1] - {1/n │n ∈ N}. Find sup(A), inf(A), min(A), max(A).

My idea of this question is to claim sup(A) and inf(A) exists (and equals a value) and prove by contradiction that min(A),max(A) exists afterwards (and equals sup(A),inf(A)). The issue that I have is ...
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0answers
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Supremum/Infimum and min/max

(Problem written verbatim) Let A = [0,1] - {$\frac1n$|n $\in$ $\mathbb{N}$}. Find supremum, infimum, maximum, and minimum of A. Justify your answers. I'm not really sure where to start. I tried ...
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2answers
22 views

Supremums and subsets

Let $A \subseteq B$, where $A$ and $B$ are non-empty sets and $B$ is bounded above. Show that $\sup(A)$ exists and $\sup(A)\le \sup(B)$ (Problem given as written) For clarity, $\sup(x)$ is the ...
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0answers
16 views

Find sum of supremum and infimum of $\tan(x)^{\sin 2x}$

Given a function $f(x) = \tan(x)^{\sin 2x}$ I have to show that it has infimum and supremum, this was easy. I already proved that $f$ takes the value equal to infimum (respectively supremum) for ...
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2answers
42 views

What exactly is the meaning of the following $\inf\{ s_n : n > N\}$ and $\sup\{ s_n : n > N\}$

What exactly is the meaning of the following $$u_N = \inf\{ s_n : n > N\} \ \ \text{ and} \ \ v_N = \sup\{ s_n : n > N\}$$ This might seem a stupid question, but I am not understanding the ...
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1answer
42 views

Existence of a sequence that converges to infimum of a function

X is a compact subset of $\Re^{n}$. A upper and lower bounded function f is defined that f:X $\rightarrow$ $\Re$. Does a sequence of {$x_n$} always exist so that f($x_n$)$\rightarrow$ $\inf$[f(x)]. ...
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1answer
81 views

Equality of sets supremum and infimum

Suppose we have a right-continuous function $f$ defined on $[0,\infty)$. I now would like to prove if $\lambda >0$ and $t\geq 0$ we have that \begin{equation} \left\{ \inf \{ s\geq 0 : f(s) \geq ...
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2answers
57 views

${\rm sup}\ A\cap B = {\rm min}\ \{ {\rm sup} (A), {\rm sup}(B) \} $

Let $A,B\subseteq \mathbb{R}$ be a non-empty intervals and bounded from above. If $A\cap B\neq \emptyset $ prove that it is bounded from above and that $Sup(A\cap B)=min\{sup(A),Sup(B)\}$ ...
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1answer
56 views

If $A$ has no max and $B$ is finite, then $\sup(A)=\sup(A\setminus B)$

Let $A\subset \mathbb{R}$ be non-empty and bounded from above, and assume it does not have a maximum. Let $B$ be a finite set of real numbers. Prove: $\sup(A)=\sup(A\setminus B)$ ...
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1answer
17 views

Prove the least upper bound of a set is in the closure of that set

If $S \subset \mathbb{R}$ and $x$ is the least upper bound of $S$, prove $x$ is in the closure of $S$. I think this means I either have to show that $x \in S$ or $x$ is a limit point of $S$, so I ...
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2answers
51 views

Prove $\sup(A)=\sup(B)$

Let there be $A,B\in \mathbb{R}$ non-empty, let assume that $\forall a\in A \exists b \in B$ such that $a\leq b$ and $\forall b\in B \exists a \in A$ such that $b\leq a$. Show that $A$ is ...
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3answers
36 views

Prove that a number $u$ is $\sup S$ given certain properties.

Problem Let $S$ be a nonempty subset of $\mathbb{R}$, and let $u$ be a number with the following properties: for each positive integer $n$, the number $u - \frac{1}{n}$ is not an upper bound of ...
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1answer
10 views

Supremums of sequences

Let $x_n$ and $y_n$ be two sequences of real numbers. Assume that $y_n$ is bounded above and that $x_n$$<$$y_n$ for all n$\in$$\mathbb{N}$. (a) Prove that $x_n$ is also bounded above. (b) Prove ...
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2answers
20 views

Proving supremums of sequences

If $l$=sup($x_n$), what is sup($kx_n$) where k$\in$$\mathbb{R}^{+}$? Prove your conjecture. I have that sup($kx_n$)=$kl$. I can prove that it is an upper bound of $kx_n$, but I'm having trouble ...
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1answer
24 views

condition for a supremum

let $A\subseteq \mathbb{R}$ be a non-empty set and $s\in \mathbb{R}$ and upper bound of $A$. So $s$ is the supremum of $A$ $\iff$ $\forall \epsilon>0$ there is $x\in A$ so $s-\epsilon<x\leq s$. ...
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2answers
52 views

Find the infimum of the set $S=\left\{\frac{1}{m}-\frac{1}{n} \, : m,n \in \mathbb{N^+}\right\}$

I need to find the infimum of the set $$S=\left\{\frac{1}{m}-\frac{1}{n} \, : m,n \in \mathbb{N^+}\right\}$$ and formally prove that it is indeed the infimum of $S$. From intuition, I know the $\inf ...
0
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1answer
33 views

Supremum/Infimum of the set $(x^2)-x<1$ where $x$ belongs to the rationals?

$$A = \{x \in \mathbb Q : (x^2)-x<1\} = \{x \in \mathbb Q : (1-\sqrt{5})/2 < x < (1+\sqrt{5})/2\}$$ Then it says therefore the supremum $= (1+\sqrt{5})/2$ and infimum $= (1-\sqrt{5})/2$. But ...
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1answer
38 views

lim inf sup integral

I have a general question about integrals of sequence of functions. Suppose $f_n \rightarrow f$ pointwise. Can I automatically say that $\lim_{n\rightarrow \infty} f_n = \lim_{n\rightarrow \infty}inf ...
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2answers
49 views

Finding $\sup$ of $\{\sqrt{n} - \lfloor\sqrt{n}\rfloor | n\in\mathbb{N}\}$

The question is to find $\sup$ and $\inf$ of $B=\{\sqrt{n} - \lfloor\sqrt{n}\rfloor | n\in\mathbb{N}\}$ where $\lfloor x \rfloor$ is defined as the largest integer that is smaller than $x$. That ...
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2answers
26 views

Prove that if $a$ is a real number with $a >2$, then there is an $n$ is an element of natural number such that $2+1/\sqrt{n}<a$

Prove that if $a$ is a real number with $a >2$, then there is an $n$ is an element of natural number such that $2+1/\sqrt{n}<a$ The goal is to show $\inf\{2+1/\sqrt{n} : ...
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1answer
26 views

Supremum vs. Maximum in the definition of the Lp norm [duplicate]

The $L_p$ norm $||A||_p$ is defined as $$\sup_{x \neq 0} \frac{||Ax||_p}{||x||_p} = \max_{||x||_p = 1} ||Ax||_p \tag{1}$$ I'm not quite getting why the LHS uses $\sup$ but the RHS uses $\max$. I ...
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1answer
39 views

Finding $\sup$ and $\inf$ of $\frac{n^5}{2^n}$ where $n$ is natural number

I'm trying to find $\sup A, \inf A$ where $$A=\{a_n=\frac{n^5}{2^n}:n\in\Bbb{N}\}, 1\not\in\Bbb{N}$$ For $n=1$ we have $a_1 = \frac{1}{2}$, $\lim_{x\rightarrow +\infty} \frac{n^5}{2^n}=0$ and after ...
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1answer
15 views

How to prove a certain inequality for a complex valued function using its derivative?

I have the following question: Let $n\in \mathbb{N}\setminus \{0\}$ be an arbitrary, but fixed natural number, let $f:\mathbb{R}^{+}\rightarrow \mathbb{C}:\ t\mapsto n^{-b} - t^{-b}$, when $b\in ...
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1answer
33 views

Is A∩B has Maximum?

Consider $A=(0,1)∪\{2\}$ and $B=(0,1)∪\{3\}$. Both sets have maxima, $2$ and $3$ respectively. But $A∩B=(0,1)$ which has no maximum. How can we say that $A∩B=(0,1)$ does not have any maximum. ...
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2answers
89 views

Prove that, for $s$ is upper bound of A, $s = \sup A$ iff , if $r < s$, so there exists $x \in A$ such that $r < x \leq s$.

Could someone verify my proof? Definition: Suppose $s \in \mathbb{R}$ and upper bounded $A \subset \mathbb{R}$. For any $x \in A$, we have $x \leq s$. For any $v$ such that $x \leq v$ for any $x$, we ...
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3answers
46 views

$\sup$ and $\inf$ of this set

These are exercises from my textbook, and I am not sure if the solutions are correct or not. Given a set $B = \{\frac{n}{2n+1} : n \in \mathbb{N} \}$ Find the $\sup$ and $\inf$ of $B$, and maxima ...
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0answers
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Determening global sup and inf points in a function

I have function: f(x)=(x+0.4)^2-cos(1+ctg(0.5x)) and I need to find global inf and sup in interval of xe[-1;1]. I tried to do the following, but with no avail: Determin domain of a function: ...
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1answer
46 views

Does the empty set have a minimum if it is a subset of R?

True or False: As a subset of $\bf{R}$: $\emptyset$ has a minimum. What is the difference between a supremum and the maximum? are they used interchangeably?
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1answer
34 views

Real analysis, boundaries question

Let $A$ be a non empty set of real numbers and let $f,g$ be functions defined on $A$ such that $f$ is bounded above on $A$ and $g$ is bounded below on $A$ . If $f(x)\le g(x) \;\forall\; x \in A$. ...
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3answers
54 views

Proof or counterexample : Supremum and infimum

If $($An$)_{n \in N}$ are sets such that each $A_n$ has a supremum and $∩_{n \in N}$$A_n$ $\neq$ $\emptyset$ , then $∩_{n \in N}$$A_n$ has a supremum. How to Prove This.
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1answer
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Supremum And infimum True or false

If the sets A and B have maxima and A ∩ B $\neq$ 0 , then A ∩ B has a maximum. Is this statement is true or false ? How to do these kind of problems
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1answer
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Are all finitely distributive and join-complete lattices infinitely distributive?

The infinite distributive law on a join-complete lattice $L$ is as follows: $\displaystyle a \wedge\left( \bigvee_{b \in B} b \right) = \bigvee_{b \in B}(a \wedge b) $ for all $a \in L$ and $B ...
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3answers
128 views

If a monotone sequence is convergent, does this imply boundedness?

I was proving a problem which states that: let $\{x_n\}_n$ be a monotone sequence of real numbers. Show that $\{x_n\}_n$ is convergent if and only if it is bounded. I have proven that if the sequence ...
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1answer
41 views

If $\sup A < \sup B$, does $B$ contain an upper bound of $A$?

If A and B are nonempty subsets of $\mathbb{R}$ and $\sup A < \sup B$, does $B$ contain an upper bound of $A$? So, the trivial case is if supB is contained in B. The Archimedean property would ...
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1answer
20 views

For every extended real number $y$ for which $y < x$, $\exists$ $n \geqslant m$ such that $y < a_{n} \leqslant x$

I have to proof this exercise for my math study, and I really don't know how to do it: Let $(a_{n})_{n=m}^\infty$ be a sequence of real numbers, and let $x$ be the extended real number $x := ...
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1answer
37 views

Absolute of a function equal to zero

How would I prove the following theorem: if $\sup \{ |f (x)| :x>0\} = 0$ this implies $|f (x)| = 0$ for all real number $x$.
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2answers
34 views

Prove if $x_n \leq k$ $\forall n \in N$ where $k$ is some constant integer, and $x_n \rightarrow x$, then $x \leq k$.

So we know that $|x_n - x| < \varepsilon$ for all $\varepsilon > 0$ for some $n \geq N$. Since $x_n \leq k$, I think we can say that $|x_n - x| \leq |k - x| < \varepsilon$ for all ...
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1answer
36 views

Prove: $-M = inf(-E)$

I have to prove this exercise for my math study: Let $E \subset \mathbb{R}$, $\mathbb{R} \ni M = sup(E)$ and $-E := ${ $-x$ | $x \in E$ }. Prove: $-M = inf(-E)$ I think I've completed the proof by ...
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1answer
20 views

basic question about infimum, minimum and countable/uncountable sets

I have not studied mathematics, so please be patient with me. Does it make sense to take the minimum over an uncountable set? In my opinion, if the set is closed in this minimum direction, then it ...
3
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1answer
36 views

$\sup A \le \beta$ proof verification

If $A \subset B, a_0 \in A$ and $\beta$ is an upper bound of B then $\sup A \le \beta.$ $\textbf{Proof:}$ Since $a_0 \in A,$ then $A \ne \emptyset,$ thus $\sup A$ exists. Since $A \subset B,$ then ...
0
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1answer
47 views

Show that $\sup \{f(x) + g(x) : x \in\ X\} \leq \sup \{f(x) : x \in\ X\} + \sup \{g(x) : x \in\ X\}$

Let $X$ be a nonempty set, and let $f$ and $g$ be defined on $X$ and ave bounded ranges in $\mathbb{R}$. Show that: $$\sup \{f(x) + g(x) : x \in\ X\} \leq \sup \{f(x) : x \in\ X\} + \sup \{g(x) : x ...
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1answer
68 views

Real Analysis Proof Continuity - lim sup anbn≤(lim sup an)(lim sup bn) and …

1) If an ≥ 0 and bn ≥ 0, prove that lim sup anbn ≤ (lim sup an)(lim sup bn) 2) If {an} and{bn} are non-negative sequences and {bn} converges, prove that lim sup anbn = (lim sup an)(lim bn). I am not ...