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

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-1
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2answers
41 views

Confusion with the reconstruction conjecture?

After reading about the reconstruction conjecture for graphs, I came up with what I thought was a proof by contradiction. Consider the class $T$ of (isomorphism classes of) finite graphs, and the ...
0
votes
3answers
73 views

How to prove a specific sequence is Fibonacci's with no prior knowledge nor trial and error?

Let $n$ be a positive integer and let $s_n$ be the number of increasing sequences of integers, alternatingly even and odd, starting with $0$ and ending with $n$. E.g. for $n=3$ we only have the two ...
2
votes
3answers
117 views

Another identity with binomial coefficients

I'm looking for an easy way to prove this identity $$\sum_{j=0}^{n}{(-1)^j j (n-j) {n \choose j}} = 0$$ for $n > 2$. I know this can be proven by differentiating $(1+x)^n = \sum x^j{n \choose j}$ ...
1
vote
0answers
32 views

Rational analogue of expansion to base b

As is well known, we can expand every positive integer $n$ to a base $b \in \Bbb N$ in the form $$n = \sum_i a_ib^i ,\ \ \ 0\leq a_i \leq b_i-1$$ uniquely. Less well known is that we can do this for ...
12
votes
1answer
325 views

Alternative proof of simple integral inequality

Problem. Let $f\in C^1(\mathbb R)$ such that $f(0) = 0$ and $0 < f'(x) \le 1$. Prove that for all $x\ge 0$ $$ \int_0^x f^3(t)\,dt \le \left(\int_0^x f(t)\,dt\right)^{\!\!2}. $$ Below is my ...
0
votes
1answer
57 views

Minimizing the area of the triangles containing a square of side $1$

This exercise is from a past admission exam to an Italian institute: Among all the triangles that contain a square of side $1$, which ones have minimum area? I have solved it, however I'd like ...
6
votes
5answers
187 views

$p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.

I want to prove that for $p,q,r$ different primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational. Is the following proof correct? If $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is rational, then ...
0
votes
0answers
34 views

(Theoritical Question) Multiply two negatives, get a positive, but… [duplicate]

Many books explain it because you can solve it with addition. I'm interested, is there another way to prove it?
1
vote
1answer
117 views

Prove that “No one likes Reggae music” is the same as “Everyone does not like Reggae music”.

I interpreted this as a case of the extension of De Morgan's Law to quantifiers. https://en.wikipedia.org/wiki/De_Morgan%27s_laws#Extensions I know that similar questions have been asked before about ...
0
votes
4answers
139 views

Proof of Pythagorean theorem without using geometry for a high school student?

There are some proofs of Pythagoras theorem which don't even require high school maths to understand, but they all are using shapes to prove of the theorem. However, I am trying to find some proofs of ...
1
vote
0answers
80 views

IMC 2014, Problem 4 [Day 2]

We say that a subset of $\mathbb{R}^{n}$ is $k$-almost contained by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points ...
3
votes
1answer
67 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 ...
3
votes
1answer
87 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} ...
2
votes
0answers
74 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 ...
7
votes
3answers
256 views

Abel-Ruffini theorem, Galois theory and minima and maxima

Questions: Does there exist a proof of the Abel-Ruffini theorem without using Galois theory? Does there exist a proof that there exists a polynomial $P$ with $\deg P = 5$ such that the roots are not ...
0
votes
2answers
75 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
67 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 $A$, ...
3
votes
4answers
196 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
54 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
140 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 ...
1
vote
2answers
44 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| ...
2
votes
3answers
128 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
1answer
27 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
54 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
2answers
156 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
36 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 ...
3
votes
3answers
192 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
91 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 ...
2
votes
0answers
61 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
76 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
41 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
140 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
67 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
279 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
239 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
76 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
85 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
33 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 ...
3
votes
2answers
171 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
54 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 ...
0
votes
1answer
51 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 ...
24
votes
1answer
603 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 ...
3
votes
2answers
212 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 ...
1
vote
0answers
48 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 ...
0
votes
0answers
65 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
46 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!} ...
1
vote
1answer
179 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
70 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
82 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
25 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 ...