If you already have a proof for some result, but want to ask for a different proof (using different methods).

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74 views

How to complete the following proof of Bolzano's Theorem?

I am trying to prove Bolzano's Theorem using the following argument. But the problem actually is that though "intuitively" I can "see" why the argument works (if I am not wrong in "seeing"), I can't ...
1
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2answers
75 views

Is an isometry necessarily surjective?

A mapping $f:X\to Y$ between metric spaces $(X,d_X)$ and $(Y,d_Y)$ is called an isometry if it preserves distances, i.e. $$d_Y(f(a), f(b))=d_X(a,b)\text{ }\forall\text{ } a,b\in X$$ My question is: ...
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2answers
33 views

$\bigcup\limits_{i=1}^n A_i$ has finite diameter for each finite $A_i$

Let $(X,d)$ be a metric space. The diameter of a set $A\subset X$ is defined to be $\operatorname{diam}(A)= \sup\{d(x,y):x,y\in A\}$. Suppose $A_1, \dots, A_n$ is a finite collection of subsets of ...
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26 views

Check my proof of showing that diam$(A)=$ diam$(\bar{A})$

Let $(X,d)$ be a metric space. The diameter of a set $A\subset X$ is defined to be diam$(A)=$ sup$\{d(x,y):x,y\in A\}$. Show that for any set $A\subset X$, diam$(A)=$ diam$(\bar{A})$ where $\bar{A}$ ...
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2answers
74 views

What is the use of Dirichlet Integral? [closed]

How can I find the value of $$\large\int_0^\infty\left(\dfrac{\sin x}x\right)^5dx$$ using Contour Integrals? I attempted it using Integration by Parts and got the an got the answer. I have studied ...
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0answers
44 views

Complex 'mean-value-theorem'-like property implies quadratic

One of my friend asked me the following problem: Problem. Suppose that $f$ is a holomorphic function on a convex open set $U$ which satisfies the following property: For all distinct $z, w \in U$, ...
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0answers
56 views

Is Doobs theorem of binary rank really true?

The theorem states that any adjacent matrix of the line graph of a connected graph has a binary rank n-1 if the order, n, of the graph is odd. I have pondered about this and found that it doesn't ...
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2answers
113 views

If all continuous functions $f: X\subset \mathbb{R}\to \mathbb{R}$ are bounded then $X$ is compact

I'm trying to show that in $\mathbb{R}$ a pseudocompact set is compact. That is, if $X\subset \mathbb{R}$ is such that all continuous functions $f: X\to \mathbb{R}$ are bounded, then $X$ is compact. ...
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0answers
68 views

If $\liminf\, |a_n|=0.$ Does there exists a subsequence of $\{a_n\}$ which has finite sum? [duplicate]

If $\liminf\, |a_n|=0.$ Does there exists a subsequence of $\{a_n\}$ which has finite sum? I tried to prove as follows: Since $\liminf\, |a_n|=0,$ then we can find $n_1<n_2<n_3\ldots$ such ...
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1answer
192 views

The quotient of a Dedekind domain by a principal ideal is a principal ideal ring.

Let $A$ be a Dedekind domain, and $a\in A-\{0\}$. I have to prove that every ideal of $A/(a)$ is principal. This is a particular case of the exercise 9.7 in Atiyah's Introduction to Commutative ...
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2answers
51 views

Proof of the Reverse Triangle Inequality

Here there is my proof (quite short and easy) of a rather straightforward result. The text of this question comes from a previous question of mine, where I ended up working on a wrong inequality. Here ...
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2answers
61 views

Proof that $|d(x,y) + d(y,z)| \leq d(x,z)$

Here there is my proof (quite short and easy) of a rather straightforward result. Still, I would like to know: if it is sound, because absolute value always creates me some problem, and if there ...
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0answers
129 views

Characteristic polynomial of adjoint

I'm trying to show that the adjoint transformation $T^*$ of the endomorphism $T$ on a finite dimensional, real inner product space has the same characteristic polynomial as $T$ in a coordinate free ...
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60 views

Symmetric proof for the probability of real roots of a quadratic with exponentially distributed parameters

What is the probability that the polynomial has real roots? asked for the probability that the quadratic polynomial $ax^2+bx+c$ has real roots if the parameters $a,b,c$ are exponentially distributed ...
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2answers
221 views

Let $f(x) = 1$ for rational numbers $x$, and $f(x)=0$ for irrational numbers. Show $f$ is discontinuous at every $x$ in $\mathbb{R}$

I am working on this proof, and wanted someone to check it and to help me understand what is happening in case (ii). The proof: Let $f(x) = 1$ for rational numbers $x$, and $f(x)=0$ for irrational ...
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1answer
67 views

Linear maps preserving the determinant and Hermiticity

Conjecture: Let $H_n$ be the space of $n\times n$ complex Hermitian matrices and let $\varphi:H_n \to H_n$ be a linear map which preserves determinants: \begin{equation} \det \circ \varphi = \det. ...
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5answers
49 views

Let $\{a_n\}$ be a sequence, $a,b$ given so that $\lim\limits_{n\rightarrow\infty} a_n=a$ and $\lim\limits_{n\rightarrow\infty} a_n=b$, then $a=b$.

Let $\{a_n\}$ be a sequence. If $a,b$ given so that $\lim\limits_{n\rightarrow\infty} a_n=a$ and $\lim\limits_{n\rightarrow\infty} a_n=b$, then $a=b$. Proof: Let $\epsilon>0$. Since ...
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0answers
17 views

Is there other ways to show reverse triangle inequality

I know that for $x,y \in \mathbb{R}$ we have that $$|x-y| \ge ||x|-|y||$$ which can be proven by writing $$|x|=|x+y+(-y)|$$ and $$|y|=|y+x+(-x)|$$ and applying triangle inequality. But I am ...
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1answer
85 views

Proof that $\lim_{x\to0}\frac{\sin x}x=1$

Is there any way to prove that $$\lim_{x\to0}\frac{\sin x}x=1$$ only multiplying both numerator and denominator by some expression? I know how to find this limit using derivatives, L'Hopital's rule, ...
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0answers
20 views

Is the following proof correct of why $\gcd(a,b)$ smallest linear combination of $a$ and $b$?

This is the proof I have: Lets see why $\gcd(a, b) $ is the smallest positive linear combination of $a$ and $b$: Let $LC = \{ s'a + t'b : s', t' \in \mathbb{Z}, s'a + t'b > 0 \}$. By the ...
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2answers
70 views

Is this a correct proof of why $\gcd(a,b) = \gcd(b, a- b)$?

I have a proof but I wasn't sure if it was correct (or how rigorous it is). I will point out what worries me. Let $D_a = \{ d : d \mid a\}$ (i.e. all elements that divide $a$) and similarly $D_b = \{ ...
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2answers
42 views

How do you show if $d \mid a$ and $d \mid b$ then $d \mid \gcd(a,b)$ without knowing that $\gcd(a,b)$ is a linear combination of $a$ and $b$?

I was trying to prove that if $d \mid a$ and $d \mid b$ then $d \mid \gcd(a,b)$ but wanted a proof that didn't require me to know that $\gcd(a,b) = ax + by$, i.e. that didn't require me to know that ...
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2answers
62 views

Is there a way to show that $\gcd(a,b) = ax + by $ without also showing that its the smallest positive linear combination?

Is there a way to show that $\gcd(a,b) = ax + by$ without also showing that it is the smallest positive linear combination? i.e. Can it be shown that there exists an $a$ and $b$ such that $\gcd(a,b) = ...
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2answers
136 views

How to show this cover of $\mathbb{Q}$ doesn't cover $\mathbb{R}$?

Let $\{q_n : n \in \mathbb{N}\}$ be an enumeration of $\mathbb{Q}$ and define $\mathcal{O} = \{I_n : n \in \mathbb{N}\}$ being $$I_n = \left(q_n - \frac{1}{2^n}, q_n + \frac{1}{2^n}\right).$$ It is ...
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5answers
50 views

Proof of sets. Need an example

My question is to show that $X-(Y \cup Z)$ is a subset of $(X-Y) \cup (X-Z)$. I already did the proof for that and understand that but the second part is to give an example to show that in general, ...
2
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1answer
102 views

Starting index of a sequence is irrelevant

"Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers, let $c$ be a real number, and let $k \geq 0$ be a non-negative integer. Show that $(a_n)_{n=m}^{\infty}$ converges to $c$ iff ...
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1answer
31 views

Solution Sets of Homogeneous Systems

I had to prove the following theorem: Suppose that $A\mathbf x=\mathbf b$ is consistent for some given $\mathbf b$, and let $\mathbf p$ be a solution. Then the solution set of $A\mathbf x=\mathbf b$ ...
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5answers
125 views

A combinatorial proof of $\forall n\in\mathbb{N},\,\binom{n}{2}=\frac{n(n-1)}{2}$

The property $\forall n\in\mathbb N,\,\binom{n}{2}=\frac{n(n-1)}{2}$ was given in our first chapter on probability theory among binomial coefficients' properties. It is really easy to prove with the ...
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3answers
140 views

Elementary theorems that require AC

It seems that AC is hiding (maybe concealed?) even in some elementary results. An example: Theorem: Let $X \subseteq \mathbb R$ and let $x_0 \in \mathbb R$ be an accumulation point of $X$. Then ...
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1answer
111 views

$T:X\to Y$ a linear operator, $Y$ finite-dimensional, then $\ker T$ is closed iff $T$ is continuous

I have to prove that if $T:X\to Y$ a linear operator on normed spaces, $Y$ finite-dimensional, then $\ker T$ is closed iff $T$ is continuous. On a lot of places I see a proof that looks like this: it ...
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1answer
39 views

Prove that $x\cdot y=0 \iff x=0$ or $y=0$ without cases

I have proved that given $x,y \in F$, $F$ a field, $x\cdot y=0 \iff x=0$ or $y=0$ by making cases for neither $x$ nor $y$ equals $0$ (and did a proof by contradiction) and then two cases for $x=0$ and ...
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1answer
35 views

Synthetic proof of a bisector length

In a exam the following was asked and I found a proof based on a law of cosines. Is there a pure synthetic proof without trigonometry? $ABC$ is a triangle with side lengths $CA=a,CB=b$. $D\in AB$ is ...
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0answers
32 views

Show that $[H,K,L] = A_5$, where $H,K,L$ have order $2$.

Let $H, K$ and $L$ be subgroups of order $2$ in some group $G$, and observe that the set $\{[h, k, l] : h \in H, k \in K, l \in L\}$ contains at most one nonidentity element, and so it generates a ...
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1answer
56 views

Showing that a wave equation $u_{tt}=ku_{xx}$ has solution $u=0$ if $u(x,0)=u_t(x,0)=0$

Show that a wave equation $\rho u_{tt}=Tu_{xx}$ has solution $u=0$ if $u(x,0)=u_t(x,0)=0$. Thoughts: This is easy using the general solution to wave equations ...
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2answers
40 views

Show that $[a,b] \subset Range(f)$

Question. Show that $[-\frac{1}{2},\frac{1}{2}]\subset Range(f)$, where $f(x)=\frac{x}{x^2+1}$. My proof. Let $y=\frac{x}{x^2+1}\Leftrightarrow yx^2-x+y=0.$ For real values of y, values of x must ...
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1answer
84 views

Contractions of maximal ideals in finite type $K$-algebras

In my commutative algebra class we proved the following theorem: Contractions of maximal ideals: Let $A\xrightarrow{\ \ f\ \ }B$ be a homomorphism of $K$-algebras. Suppose that $B$ is of finite ...
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0answers
17 views

Other proofs of uniqueness of interpolating polynomial

I think that one of the well known proofs is this one: Let $f:[a,b]\to\mathbb{R}$ be a function and $P_n:[a,b]\to\mathbb{R}$ be the interpolating polynomial for $f$ on $[a,b]$. Let the nodes of ...
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1answer
89 views

The value of $\gcd(2^n-1, 2^m+1)$ for $m < n$

I've seen this fact stated (or alluded to) in various places, but never proved: Let $n$ be a positive integer, let $m \in \{1,2,...,n-1\}$. Then $$\gcd(2^n-1, 2^m+1) = \begin{cases} 1 ...
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3answers
38 views

Proof of $\lim_{k\to \infty}{|x^k-x|\over { 1+|x^k-x|}}=0$

Is there an easy way to prove that $$\lim_{k\to \infty}{|x^k-x|\over { 1+|x^k-x|}}=0$$ with $x\in \mathbb R$ without using $\epsilon-\delta$ definition? Any help would be really appreciated
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1answer
36 views

Different ways of Proving the existence of Tensor Product

This is Just a curosity. Let $A$ be a commutative ring and $M,N,P$ be A-modules.I know that tensor product of $M$ and $N$ is a universal object ($ M \otimes N$,u) (where $M \otimes N$ is a $A$-module ...
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0answers
66 views

(ZF - Foundation) proves that $\log_2$ can't be infinitely iterated: Alternative proof

I think I solved Ex. (12) in Chapter I of Kunen's book. It states that ZF sans Foundation proves: For every set $X$, $$ \aleph(X) < \aleph(\mathcal{P}^3(X)), $$ where $\aleph(X):= \sup\{\alpha \in ...
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1answer
192 views

Alternative proof of the Riemann Sum Theorem using Mean Value Theorem for Integrals.

I've been reviewing proofs for a couple of calculus theorems and as I was trying to recall the proof of the Riemann Sum Theorem which uses Lower Sums and Upper Sums I came up with an idea to prove it ...
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1answer
30 views

Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one.

Let $X,Y$ be nonempty sets and $f:X\to Y$ be a function. Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one. My approach is by ...
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2answers
27 views

Clarification: Prove there exists a number $N$ such that $n > N$ implies $s_n >a$

Below is the proof that I have been working on and the solution provided by the professor. Let $(s_n)$ be a convergent sequence, and suppose $\lim s_n > a$. Prove there exists a number $N$ ...
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1answer
23 views

Find a criterion for divisibility

Find a criterion such that $\displaystyle\sum_{i=1}^ni$ divides $\displaystyle\prod_{i=1}^ni^2$ for $n\in\mathbb N$. What I have done so far, $\displaystyle\sum_{i=1}^ni=\frac{n(n+1)}{2}$ and ...
5
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1answer
68 views

Proving that and how $ \frac{1}{n}\sum\limits_{p\le n}\lfloor n/p \rfloor - \sum\limits_{p\le n} 1/p $ approaches $0$

Let $p$ denote a generic prime number. By Mertens' second theorem, the sequence $$\sum\limits_{\ p \le n} \frac1p - \log\log n$$ converges to the Meissel-Mertens constant $M\approx 0.2614972$. Now let ...
2
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1answer
46 views

Cantor's diagonal argument modified version

I have the following doubt regarding Cantor's diagonal argument. First of all, the "usual case" is quite clear for me. If $X$ is some set, then we can show there is no surjection from $X$ onto the set ...
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2answers
74 views

Prove that no set can contain everything (or every other set)

Prove that there cannot exist a set that contains everything. Ill put my proof in the answer so please check it there. Also if there is a more creative way to do this(using the basic axioms) if it's ...
1
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1answer
36 views

An alternative method to find $\sum_{k=1}^{2n-1} | \beta ^k - 1|$

Let $\beta \in \mathbb{C}$ such that $\beta ^n = 1$ but $ \beta ^k \neq 1$, $\forall k=1,2,\cdots, n-1$. Find the value of $$ \sum_{k=1}^{2n-1} | \beta ^k - 1|.$$ I came across such a question ...
0
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1answer
74 views

How to prove triangle inequality in How to Prove It Sec. 3.5 Question 12c?

(a) Prove that for all real numbers $a$ and $b$, $$|a| \le b \text{ iff } -b \le a \le b.$$ (b) Prove that for any real number $x$, $$-|x| \le x \le |x|.$$ (Hint: Use part (a).) (c) Prove that ...