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

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4
votes
7answers
92 views

Show an $\arctan$ and $\arcsin$ function is constant

Show that for every $x\geq1$ the following is true: $2\arctan x + \arcsin \frac{2x}{1+x^2} = \pi$ One way (mentioned in the link at the bottom) would be to calculate the derivative of the left side, ...
8
votes
2answers
143 views

Proving $\int_{0}^{\pi/2}x\sqrt{\tan{x}}\log{\sin{x}}\,\mathrm dx=-\frac{\pi\sqrt{2}}{48}(\pi^2+12\pi \log{2}+24\log^2{2}) $

When trying to solve this problem: How to Integrate $ \int^{\pi/2}_{0} x \ln(\cos x) \sqrt{\tan x}\,dx$ I found his sister integral has an interesting closed form provided my calculation is correct. ...
2
votes
2answers
70 views

$\frac{1}{{9\choose r}} -\frac{1}{{10\choose r}} = \frac{11}{6{11\choose r}}$. Is there a way to find $r$ without using algebra?

$$\frac{1}{\dbinom 9r} -\frac{1}{{\dbinom{10}r}} = \frac{11}{6\times \dbinom{11}r}$$ I guess directly applying algebra for this problem would be enough. But are there any simpler and prettier ...
2
votes
0answers
58 views

Proof Verification for Putnam Problem [Alternate Solution] 1997 A4

I have come across an interesting problem from the Putnam 1997 test, question A4: Problem: Let $G$ be a group with identity $e$ and $\phi: G \to G$ a mapping such that $\phi(g_1)\phi(g_2)\phi(g_3) = ...
4
votes
2answers
41 views

Proving a formula using another formula

These questions are from the book "What is Mathematics": Prove formula 1: $$1 + 3^2 + \cdots + (2n+1)^2 = \frac{(n+1)(2n+1)(2n+3)}{3}$$ formula 2: $$1^3 + 3^3 + \cdots + (2n+1)^3 = ...
2
votes
0answers
33 views

Clever proofs that prove one identity is equal to another, without going through the original identity?

In a previous question I attempted to formalize the argument of going from one proof of an identity to another, which turned out to be harder than I thought. The thing is, while it may be impossible ...
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 ...
1
vote
1answer
58 views

Did I solve exercise 4.5.4 (b) of 'How to Prove it' by velleman correctly and concisely?

4.5.4 Suppose R is a strict partial order on A. Let S be the reflexive closure of R. (b) Show that if R is a strict total order, then S is a total order. Suppose R is a strict total order. ...
1
vote
1answer
64 views

Finding all solutions to $x^2+y^2=2010$

I need to find all integer solutions to $x^2+y^2=2010$. we can take $x\leq y$ for commodity. The problem can be tackled through brute force. We need $1005\leq x^2\leq 2010$ and so $32\leq x \leq ...
1
vote
2answers
31 views

Extreme value theorem, without Heine Borel.

I was wondering, if there are any mistakes, in this proof of the extreme value theorem: Theorem. Let $X$ be a compact set and $f:X\rightarrow\mathbb{R}$, s.t. $f$ is continuous. Then there exists ...
6
votes
1answer
118 views

Seeking a More Elegant Proof to an Expectation Inequality

Let $X$ and $Y$ be i.i.d. random variables, and $\mathbb E[|X|]<\infty$, prove that $$\mathbb E[|X+Y|]\geq\mathbb E[|X-Y|].$$ This question is a re-posting of An expectation inequality. I can ...
0
votes
0answers
42 views

To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.

There's again one small detail on which I'm not sure. (Proposition 44 - book 1) http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI44.html Here's the quote : "Then HLKF is a parallelogram, HK ...
0
votes
0answers
24 views

Triangles which are on the same base and in the same parallels equal one another.

I have a small question regarding proposition 37 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI37.html The only problem I got with the proof is the fact that we ...
0
votes
0answers
18 views

Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel

I have a small question regarding proposition 33 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI33.html We want to prove that two lines joining equal parallels ...
3
votes
2answers
81 views

Theorem 7.2 in General Topology by S. Willard

Theorem 7.2 If $X$ and $Y$ are topological spaces and $f:X \to Y$ , then the following are all equivalent :- I) $f$ is continuous. II) for each E $\subset X$ , $f(\bar E) \subset ...
2
votes
0answers
72 views

Union of infinite broom and topologist's sine, connectednes, locally connectednes properties…

I'd like to know if my answer of the following exercise is correct. I really appreciate any suggestion you can provide to improve my argument or corrections in case I made a mistake :) Let ...
1
vote
1answer
43 views

A question on Primes in Arithmetic Progression

We know that an arithmetic progression has to have a composite number since there are arbitrarily large gaps between primes. But I was wondering whether the following construction is possible: Can ...
30
votes
1answer
537 views

Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?

In a gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$: Proof of equality ...
1
vote
1answer
65 views

Question 7F from general topology by Stephen willard?

Can someone help me with 7F from Willard? In part two : $\mathbf{7}$F. Functions to and from the plane. The facts presented here for the plane will be proved in more generality for ...
0
votes
0answers
47 views

$\sum a_n$ converges $\implies\ \sum \sqrt{a_na_{n+1}}$ converges?

Let $a_n > 0.$ When $\sum a_n$ converges $\sum \sqrt{a_n a_{n+1}}$ converges or not? For, $$\frac{\sqrt{a_n a_{n+1}}}{a_n}=\frac {\sqrt{a_{n+1}}} {\sqrt{a_n}}$$ $\because$ By comparison test ...
3
votes
0answers
96 views

Should a certain entire function be a polynomial?

Assume $f$ is an entire function such that $$\lim_{z\to\infty}\frac{|f'(z)|}{1+|f(z)|^2}=0,$$ then should $f$ be a polynomial? Picard's Theorem proves this instantly; which states: Let $f$ be a ...
0
votes
0answers
62 views

Finite solution of Power Diophantione Equation.

Given an equation $x^2+k=y^3$ where k is a constant and $y=f(x)$,$f(x)$ is differentiable and algebraic. for which- $$\frac{d}{dx}x^{2} \neq\frac{d}{dx} f(x)^3$$ 1. Can I infer that the ...
1
vote
1answer
54 views

Proving that a punctured disk is not simply connected, using a specific definition

I am dealing with the same set based on my previous question. I want to show that the set $H = \{z \in \mathbb{C} : 0 < |z| < 1\}$ is NOT simply connected, using the following definitions, that ...
2
votes
2answers
87 views

Proofs of theorems, where picture is sufficient

A while ago I have had the pleasure to come across those lectures of Topology & Geometry by Dr Tadashi Tokieda (I do recommend watching at least the first lecture, both parts). My question is ...
2
votes
2answers
87 views

Is $x^4+nx+1$ irreducible?

Consider the polynomial $\xi= x^4+nx+1\in \mathbb Z[x]$. Show that if $n=\pm2$ then $\xi$ is reducible and that $n\neq\pm2$ implies $\xi$ is irreducible. I got the answer by writing the ...
4
votes
1answer
42 views

Proving the uniqueness of the weak limit

In "A First Look at Rigorous Probability Theory" by J. S. Rosenthal there is the following exercise: Prove that weak limits, if they exist are unique. That is, if $\mu, \nu, \mu_1, \mu_2, \ldots$ ...
5
votes
3answers
87 views

Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?

Let $G$ be a finite group, and $g$ be a an element of $G$. How could we go about proving $g^{|G|}=e$ without using cosets? I would admit Lagrange's theorem if a proof without talking about cosets can ...
16
votes
1answer
108 views

If the Greeks had been four dimensional, would they have been able to derive the pi squared coefficient for the hypersphere volume without calculus?

I was reading about Archimedes' pre-calculus proof of the volume of the sphere and I realized that the trick he uses (volume of hemisphere + volume of cone = volume of cylinder) doesn't generalize to ...
1
vote
2answers
53 views

Seemingly easy Ordinary Differential Equation

For which values of $T$ can we find a unique solution of the ODE $x''(t) = −x(t) $ satisfying the boundary conditions $x(0) = a_1$ and $x(T) = a_2$ for any values of $a_1$ and $a_2$ ? I can solve ...
2
votes
0answers
40 views

Let $p\gt 3$ be a prime number and: $\sum_{j=1}^{p-1}\frac{(-1)^{j}}{j} \binom{p-1}{j} =\frac{a}{b}\Rightarrow p^2\mid a$

I want to prove the following statement: Let $p\gt 3$ be a prime number and let: $$\sum_{j=1}^{p-1}\frac{(-1)^{j}}{j} \binom{p-1}{j} =\frac{a}{b}$$ Which $a,b\in \mathbb Z$ and $\gcd(a,b)=1$. ...
0
votes
2answers
58 views

On a connection between Newton's binomial theorem and general Leibniz rule using a new method.

In calculus the general Leibniz rule asserts that Let $n$ be a natural numbers, if $f$ and $g$ are $n$-times differentiable functions at a point $x$, then the function $fg$ is also $n$-times ...
0
votes
1answer
51 views

Show $f(x)$ is bounded in a neighbourhood of its limit points

The attempt I made doesn't cover the case for $x=c$. How can I make it so it does? Prove that if a function $f : A \to \mathbb{R} $ has a limit $l \in \mathbb{R} $ at $c \in L(A)$, then it is bounded ...
4
votes
1answer
67 views

Is the solution of functional equation $x^x=y^y$ when $0\lt x\lt y$ uncountable?

I want to prove that, the set: $S=\{(x,y)\in \mathbb R^2\,\,|\,\,0\lt x\lt y \,\,,\,\,\,\,x^x=y^y \}$ $\,\,$ is uncountable. My idea is the following: Consider the function ...
0
votes
1answer
8 views

To proof the difference images is a subset of their map difference sets

Let $A$ and $B$ sets, with $P,Q \subseteq A$ and let $f:A \to B$ 1) prove that $f(P)-f(Q) \subseteq f(P-Q)$ 2)Is it necessarily the case that $f(P-Q) \subseteq f(P)-f(Q)$? Give a proof or a ...
0
votes
0answers
39 views

proof that $\frac{e^{-t}}{2}(t^2+2t+2)\le1$ for $t\ge0$

Show that $\forall t\ge0,x\le1$ where $$x=\frac{e^{-t}}{2}(t^2+2t+2),t\in\mathbb{R}.$$ My proof we have $x=\frac{(t^2+2t+2)e^{-t}}{2}$ then ...
9
votes
3answers
1k views

Prove that there is no smallest positive real number

I have to prove the following: $$\text{Prove that there is no smallest positive real number}$$ Argument by contradiction Suppose there is a smallest positive real number. Let $x$ be the smallest ...
2
votes
0answers
49 views

Product Spaces: Tube Lemma

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. My professor asked to prove the following Lemma. The Tube Lemma: Let $K$ be a compact ...
1
vote
1answer
69 views

Dijkstra's Algorithm for Negative Weights.

Now the problem states that their is a graph $ G = (V,E) $ where some of the edges have negative weights while some of the edges have positive edges. Now the question is why won't Dijkstra's algorithm ...
3
votes
3answers
103 views

Can I find $\sum_{k=1}^n\frac{1}{k(k+1)}$ without using telescoping series method?

I am interested to find $$\sum_{k=1}^n\frac{1}{k(k+1)}$$ without using telescoping series method. I tried very hard but still could not think of a way to find it without using telescoping series ...
0
votes
0answers
35 views

Proving associativity of symmetric set difference

I'm proving that $P(X)$ (the set of the subsets of $X$) is a ring with the following operations: If $A, B \subset X$, then $A+B := (A \cup B) \backslash (A \cap B) $ and $A \cdot B = A \cap B $. I ...
5
votes
1answer
78 views

$\mathbb{Z}[\sqrt{2}]/(3-\sqrt{2})$ is ring isomorphic to $\mathbb{Z}_n$.

what would be an $n$ such that $\mathbb{Z}[\sqrt{2}]/(3-\sqrt{2})$ is ring isomorphic to $\mathbb{Z}_n$? This problem was on a qualification test. Here's how I solved it, but I'm not satisfied ...
0
votes
0answers
36 views

Lebesgue Measurable By Alternative definition of Measure

Prove that any compact set $K$ in $R^{n}$ is Lebesgue measurable and $m(K) < \infty$ Actually the proof of this is given in Stein and Shakarchi's book on Real Analysis (Page 38, Property 4) where ...
2
votes
1answer
70 views

Where can I find Wielandt's original proof of Sylow's Theorem?

I have seen several proofs of Sylow's Theorem based on Wielandt's method. Everyone gives credit to Wielandt's proof of Sylow's theorem, but ironically everyone puts their own spin on it. Where can I ...
2
votes
8answers
356 views

A pedagogical proof that 9's can be ignored when calculating digital roots

I was asked by an elementary school teacher for a proof that you can ignore all 9's when calculating the digital root of a number. For instance, when calculating the digital root of 7593329, you ...
2
votes
2answers
66 views

Proof of irrationality without using contradiction

I'm just wondering if there exists proofs that certain numbers are irrational that do not begin by saying some like along the lines of "assume $k=a/b$ for integers $a$ and $b$" and then deduce a ...
2
votes
0answers
74 views

How can one prove the existence and uniqueness of solutions to linear differential equations?

It is a theorem (I think) that the equation: $$\mathbf{x}'(t) = A(t)\mathbf{x}(t) + \mathbf{b}(t); \qquad \qquad \mathbf{x}(t_0) = \mathbf{x}_0$$ Has a unique global solution for any matrix ...
2
votes
2answers
92 views

Proving $2^{2^n}+3^{2^n}+5^{2^n}$ is divisible by $19$ for all $n\geq 1$ by induction

I came across the following in the book Handbook of Mathematical Induction: $$ 19\mid (2^{2^n}+3^{2^n}+5^{2^n}),\quad n\in\mathbb{Z^+}\tag{1} $$ Apparently, this problem is not so bad if you think ...
0
votes
4answers
56 views

If $x>y$, then $x \bmod y < \frac{x}{2}$

Given two natural numbers $x$ and $y$ such that $x > y$, prove that $$x \bmod y < \frac{x}{2}.$$ I was planning on solving this proof by using the definition of mod; however, I was wondering ...
0
votes
0answers
22 views

Proof Method and Absolute Value

I'm interested in the following method of proof for inequalities involving modulus : $( -x \geq C \wedge x \geq C ) \to ( |x| \geq C )$ $( -x > C \wedge x > C ) \to ( |x| ...
2
votes
2answers
81 views

Is there a much simpler proof for Euler factorial formula?

Euler formula for factorial stated as follows Theorem [Euler]: For any non negative integers $a$ and $n$ such that $a\geq n$ $$ n!=\sum_{k=0}^{n}(-1)^k\binom{n}{k}(a-k)^n$$ Proving this ...