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

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2
votes
1answer
51 views

How to resolve the issue of two sequences converging to zero for $n, m \to \infty$?

My question is motivated by the following exercise in probability theory: Let $X_n \to X$ in probability and $X_n \geq Y$ a.s. Show that $X \geq Y$ a.s. I noticed that for all $n, m \in ...
7
votes
1answer
85 views

Largest rectangle bounded under a function

Let $f$ be a positive monotonically increasing real function in $[0,1]$. Let $F$ be the area under the curve of $f$ ($F=\int_0^1{f(x)dx}$) For every $x\in[0,1]$, let $G(x)=f(x)*(1-x)$ = the area of a ...
3
votes
0answers
64 views
+50

Minima and maxima of the 6th degree polynomial are not expressible in radicals.

Question: Prove that there exists a polynomial $P$ with $\deg P \geq 6$ such that the minima and maxima are not expressible in radicals. I have the following proof: the minima and maxima of a 6th ...
1
vote
5answers
63 views

Concise induction step for proving $\sum_{i=1}^n i^3 = \left( \sum_{i=1}^ni\right)^2$

I recently got a book on number theory and am working through some of the basic proofs. I was able to prove that $$\sum_{i=1}^n i^3 = \left( \sum_{i=1}^ni\right)^2$$ with the help of the identity ...
3
votes
1answer
78 views

Solving the trigonometric equation $\tan^2x+\cot^2x=2-\cos^{2014}(2x)$

I was solving the trigonometric equation $$\tan^2x+\cot^2x=2-\cos^{2014}(2x) $$ I solve it by inequality $|a|+\frac{1}{|a| }\geq 2$. $$ L.H.S=\tan^2x+\cot^2x =\tan^2x+\frac{1}{\tan^2x} ...
1
vote
0answers
21 views

Hypercenter is the intersection of normalizers of Sylow subgroups.

I'm trying to prove that the intersection of the normalizers of the Sylow subgroups of a [finite] group $G$ is equal to its hypercenter, i.e., $$Z_\infty(G)=\bigcap\limits_{S\in ...
0
votes
2answers
42 views

Use proof by contradiction to prove that if $a$ and $b$ are odd integers, then $4 \nmid (a^2+b^2)$.

I have been working through the following proof: Use proof by contradiction to prove that if $a$ and $b$ are odd integers, then $4 \nmid (a^2+b^2)$. Below, I have included screenshots of the ...
0
votes
3answers
59 views

Help with proof: $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$

The question is: If $A,B$ are any $m\times n$ matrices, prove that $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$. ($\mathrm{rank}(A)$ is the dimension of the column space of $A$, ...
30
votes
12answers
646 views

What are the theorems in mathematics which can be proved using completely different ideas?

I would like to know about theorems which can give different proofs using completely different techniques. Motivation: When I read from the book Proof from the Book, I saw there were many ...
3
votes
4answers
99 views

$S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$.

Here is the problem that I am currently working on: $S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$. I have access to the answer for this proof, and wanted help with the first ...
2
votes
2answers
45 views

What exactly is $\cap$-stable here?

From my lecture notes: Theorem: Let $(\Omega, \mathcal A, P)$ be a probability space, $A \in \mathcal A, \mathcal M := \{ M_1, \ldots, M_n \} \subset \mathcal A$. The following statements are ...
0
votes
2answers
54 views

Prove that if $a<1/a<b<1/b$ then $a<-1$

The following is Exercise 3.2.8 from Velleman: Suppose that $a$ and $b$ are nonzero real numbers. Prove that if $a<1/a<b<1/b$ then $a<-1$. I solved it using the hint in the back of ...
2
votes
3answers
64 views

Prove that $\sum_{d|n}\phi(d)=n$ where $\phi$ is the Euler's phi function, $n,c\in\mathbb{N}$

Here is a very elementary number theory proof using strong induction. Please mark/grade. Prove that $$\sum_{d|n}\phi(d)=n$$where $\phi$ is the Euler's phi function, $n,d\in\mathbb{N}$ First, ...
1
vote
2answers
14 views

A finite set and the set of its fixed points under any involution have cardinalities of the same parity

I am trying to write down a formal proof of the following fact: Let $A$ be a non-empty finite set and $f$ an involution on $A$. If $A'$ is the set of fixed points of the involution $f$, then $|A| ...
1
vote
1answer
23 views

Asymptote criterion

Let $f:(a, \infty)\to \Bbb R$ be a differentiable function such that exists $\lim_{x\to\infty}f(x)=l<\infty$ and exists (in the sense it can also be infinity) $\lim_{x\to\infty}f'(x)$. Under these ...
0
votes
1answer
39 views

Prove that $H\times K \cong K\times H$

According to the book: Let $G$ be the internal direct product of subgroups $H$ and $K$. Then $G$ is isomorphic to $H\times K$. From that it results $H\times K \cong K\times H$. Is there any ...
8
votes
12answers
1k views

How to prove that $\lim\limits_{x\to0}\frac{\tan x}x=1$?

How to prove that $$\lim\limits_{x\to0}\frac{\tan x}x=1?$$ I'm looking for a method besides L'Hospital's rule.
8
votes
2answers
124 views

Why is periodic harmonic analysis only possible with sines?

This paper shows that if we consider odd functions on $(-\pi,\pi)$ in $L_2$, then the only $2\pi$-periodic function $f$ for which $f(nx)$ is a complete orthogonal system is the sine function. I'll ...
0
votes
0answers
27 views

Is my proof of 'inscribed angle theorem' different from the usual one?

The usual proof of inscribed circle theorem uses the fact that the sum of interior angles is equal to exterior angle. I've formulated a little different approach, although the underlying concept is ...
51
votes
15answers
1k views

What is the most unusual proof you know that $\sqrt{2}$ is irrational?

What is the most unusual proof you know that $\sqrt{2}$ is irrational? Here is my favorite: Theorem: $\sqrt{2}$ is irrational. Proof: $3^2-2\cdot 2^2 = 1$. (That's it) That is a ...
4
votes
1answer
245 views

Heron's Formula; an Intuitive or Visual Proof

I've found several proofs for Heron's formula for the area of a triangle in term of its sides, but none of them is simple and intuitive enough to show WHY the formula works. Do you know an intuitive ...
0
votes
1answer
953 views

Proving graph connectedness given the minimum degree of all vertices

I know that this is a repeat of a previous question asked with a similar title, but I didn't want to revive an old thread. The solution presented in that thread seems to be the common one, but I was ...
3
votes
3answers
167 views

Possible alternative way of expressing continuity of a function?

In Calculus or Real Analysis the usual form of definition of continuity of a function is $\epsilon- \delta$ def. From a rigorous point of view, is it possible to say this way? and if so, why?: ...
2
votes
2answers
35 views

Expected value of cube projection

This problem is from the book of V. I. Arnold. Find the expected area of the projection of a unit cube onto the plane under isotropic random direction of projection. The direct evaluation of it ...
0
votes
2answers
119 views

Proof of Lindelöf Theorem

I have been surfing the net to read the proof of the Lindelöf Theorem: Let $U\in \mathbb{R}^n$ be open and $U=\bigcup_{\lambda \in \Lambda} U_{\lambda}$where $\Lambda$ is an index set, ...
2
votes
0answers
49 views

proof about triangles with sides in arithmetic progression

$\require{cancel}$ The sides of a triangle are in an arithmetic progression with $k>0$ and $a>b>c$. Prove that ...
0
votes
1answer
49 views

How to prove the External Bisector Theorem by dropping perpendiculars from a triangle's vertices?

I've found two different methods to prove Internal Angle Bisector Theorem, viz. Wikipedia ("Proof 2") method and AskMath.com method. How can we prove External Angle Bisector Theorem with ...
1
vote
1answer
35 views

Exercise about nbd-finiteness (Dugunji III.9.1)

Sorry for the vague title, but the question is fairly long: Let $\{A_\alpha\}$ be a ndb-finite closed cover of $X$. Consider $x_0\in X$, and let $A_{\lambda_i}$ be (all) the $A_\alpha$ that ...
0
votes
5answers
124 views

Different proofs that $\lim_{n\to\infty}\sin n$ does not exist [duplicate]

In this question it was proved that limit $$ \lim_{x\to\infty}\sin x $$ doesn't exists. What about $$ \lim_{n\to\infty}\sin n? $$ I asking about usual limit, where $n$ is integer. I know that this ...
0
votes
0answers
58 views

Easy proof of Cayley Hamilton theorem [duplicate]

What is wrong with this proof of Cayley hamilton? If $A$ is $n \times n$ matrix and $P$ is its characteristic polynomial then $P(A) = 0.$ Proof: $P(x) = \det(A - xI) \implies P(A) = \det(A - A) = ...
3
votes
4answers
250 views

On the value of proofs vs counterexamples

If a conjecture doesn't hold we usually provide a counterexample. While re-proving theorems is valuable and mathematicians do it usually, I think proving that some statement is wrong without giving ...
7
votes
4answers
212 views

Looking for a direct proof of the following exercise

A friend of mine told me about the following problem: Let $\{r_n\}$ be a sequence of rational numbers such that $\lim_{n\to\infty}r_n=x\in\Bbb R,$ $r_n\neq x,$ for every $n\in\Bbb N$ and ...
1
vote
2answers
53 views

What is your favorite proof of the Pythagorean Theorem? Why? [duplicate]

My favorite is Euclid's original proof for two reasons: First, it requires minimal raw material. It only needs the result that the area of a triangle is half the area of a rectangle with the same ...
3
votes
0answers
58 views

Are there less trivial necessary and sufficient conditions?

Given an infinite set $X$ with the finite-complement topology, find a necessary and sufficient condition for a map $f:X\to X$ to be continuous. I came up with the condition that $\lvert ...
2
votes
0answers
28 views

The proof of the integral test using the contradiction method.

I am currently writing a short note about the proof techniques. I found a random theorem and wanted to write a proof by contradiction as an example. The theorem says The integral ...
19
votes
1answer
290 views

Does Nakayama Lemma imply Cayley-Hamilton Theorem?

Consider the Cayley-Hamilton Theorem in the following form: CH: Let $A$ be a commutative ring, $\mathfrak{a}$ an ideal of $A$, $M$ a finitely generated $A$-module, $\phi$ an $A$-module endomorphism ...
0
votes
1answer
53 views

If $\{a_{n_k}\}$ is a subset of $\{a_n\}$, $\lim_{k\to\infty} a_{n_k }= \lim_{n\to\infty} a_n\ $

Let $\{a_n\}$ be a sequence and L a real number such that $\lim_{n\to\infty} a_n = L$ Prove that if $\{a_{n_k}\}$ is any subsequence of $\{a_n\}$, then $\lim_{k\to\infty} a_{n_k} = L $ I have ...
2
votes
0answers
60 views

Euler-Mascheroni constant [strategic proof]

I know two proof about the approximation of Euler-Mascheroni constant $\gamma$, but very technical. So I would like to know if someone has a strategic proof to show that $0,5<\gamma< 0,6.$ ...
0
votes
1answer
35 views

Prove that $\{a_n\}$ is bounded [duplicate]

Let $\{a_n\}$ be a sequence an $L$ a real number such that $\lim_{n\to\infty} a_n = L$ Prove that $\{a_n\}$ is bounded This reminds me of the bounded monotone convergence theorem (BMCT) but in ...
5
votes
2answers
120 views

Determine the number of subsets

How many distinct subsets of a set $\text{A}$ are there, containing at least $9$ elements, where the total number of elements in set $\text{A}$ is $18$ ? I've solved it by making cases of either ...
2
votes
0answers
45 views

A better way to answer this question

So my team and i were asked this question a few years ago on a small Math-A-Thon on my hometown. It went something like this: "We need to transport a neon tube (or any tube, who cares) of 92cm ...
1
vote
1answer
120 views

Change of variable formula for the image of a hypercube

Let $\varphi: \mathbb{R}^n\to \mathbb{R}^n$ be an injective $C^1$ map. Let $I=[0, 1]^n$. I want to show that $$m(\varphi(I))=\int_I \left|\det D\varphi(x)\right|dx.$$ This is a special case of the ...
5
votes
1answer
723 views

Characterization of positive definite matrix with principal minors

A symmetric matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$. However, such matrices can also be characterized by the positivity of the principal minors. A statement and proof can, ...
0
votes
0answers
56 views

Continuous function $0$ on one closed set and $1$ on the other

Looking for a better approach of the following question if possible. Question: Let $A$ and $B$ be disjoint nonempty closed sets in a metric spaces $X$, and define ...
1
vote
1answer
45 views

Searching simpler proof for convergence of a sequence

It is known that if $f_n \to f$ uniformly and $x_n \to x$ then $f_n(x_n) \to f(x)$. As an example, this can be applied in order to show that $$\sum_{k=0}^n \frac{\left( 1-\frac{1}{n} \right)^k}{k!} ...
49
votes
1answer
2k views

$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?

If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and ...
0
votes
1answer
85 views

Hamel Basis in Infinite dimensional Banach Space without Baire Category Theorem

Prove that every Hamel basis in an infinite dimensional Banach Space is uncountable without using Baire Category Theory. We are assuming axiom that vector space dimension (if exist) is well-defined. ...
2
votes
2answers
52 views

Alternative Quadratic Formula

Well the formula for solving a Quadratic equation is : $$\text{If }\space ax^2+bx+c=0$$ then $$x=\dfrac{-b \pm \sqrt{b^2 -4ac} }{2a}$$ But looking at this : [Wolfram Mathworld] (And also in other ...
1
vote
1answer
48 views

Prove that the image of a a closed and bounded interval in $\mathbb{R}$ is a a closed and bounded interval in $\mathbb{R}$?

According to the excercise 7.22 of the book Topology by Franzosa: Combining the Extreme Value Theorem and the Intermediate Value Theorem, prove the following theorem: Let $[a, b]$ be a ...
2
votes
3answers
17 views

Direct Proof for Statement on Linear Independence and Unique Representations

The Statement Show that if a set of vectors is linearly independent, then any vector in the span of that set has a unique representation as a linear combination of these vectors. My Proof I'm going ...