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

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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
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1answer
108 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 ...
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0answers
24 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 ...
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23 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 ...
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14 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 ...
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2answers
71 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 ...
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62 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 ...
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1answer
39 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 ...
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1answer
410 views
+100

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 ...
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1answer
54 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 ...
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46 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 ...
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92 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 ...
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0answers
54 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 ...
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1answer
40 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
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2answers
72 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 ...
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2answers
85 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 ...
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1answer
36 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 $\mu, \nu, \mu_1, \mu_2, \ldots$ ...
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3answers
84 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 ...
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1answer
90 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 ...
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2answers
51 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 ...
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0answers
38 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 $\gcd(a,b)=1$. ...
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2answers
43 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 ...
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1answer
48 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
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1answer
64 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 ...
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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 ...
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0answers
37 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 ...
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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
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0answers
33 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 ...
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52 views

Determining if a rational number has a terminating decimal expansion (proof)

Theorem: $x=\frac pq$ is any given rational number, $n$ and $m$ are any whole numbers (including zero) which you can choose. a) If $q=2^n5^m$ is possible, $x$ has a terminating decimal expansion. ...
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1answer
52 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
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3answers
99 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 ...
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0answers
29 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 ...
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1answer
60 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 ...
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0answers
26 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
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1answer
64 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
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8answers
347 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
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2answers
64 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
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0answers
63 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
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2answers
89 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 ...
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4answers
54 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 ...
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0answers
21 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| ...
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2answers
73 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 ...
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2answers
49 views

Proof that real multiplication distributes over addition using Dedekind cuts?

Proving $\forall x,y,z\in \mathbb R,\:x(y+z)=xy+xz$ There is a very concise proof of this in ProofWiki using Cauchy sequences, but I was wondering whether the same would be possible using Dedekind ...
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2answers
47 views

determine the values that series converges

Determine for what values of $x \in \Bbb R$ the series $$\sum_{n = 1}^\infty \frac{(-1)^n}{2n+1}\left(\frac{1-x}{1+x}\right)^n$$ coverges. I have tried the alternating series test but I don't think ...
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0answers
27 views

How to show that a function is continuous in the topology of weak convergence

Let $\Omega$ be compact, and let $\omega^* \in \Omega$ be arbitrary. Let $\Delta (\Omega)$ denote the set of all probability measures over $\Omega$, and endow $\Delta ( \Omega)$ with the topology of ...
3
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1answer
144 views

Is there a memorable solution to Kirkman's School Girl Problem?

Given a solution to Kirkman's School Girl Problem, it is of course easy enough to check that it actually is a solution. But how could you reconstruct it if you lost it? Is there a method or algorithm ...
2
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1answer
61 views

Geometric proof of this property of the ellipse

I came across the following property of the ellipse: The distance from a focus of an ellipse to any point on the ellipse is equal to $a(1-e \cos\theta)$. Where the $a$ is the length of ...
2
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1answer
29 views

adjoint of an operator. on $L^2(0,1)$, $Bf(x)=\int_0^x f(t)dt$

I see that the above operator is bounded. I ended up with an argument to calculate the adjoint as follows, $$ <f,Bg>=\int_0^1\overline{f(x)} \int_0^xg(t)\,dt\,dx $$ I see $f(x)$ as the ...
2
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1answer
29 views

Consider a linear operator $L$ and some polynomial of it, $L'=p(L)$. Show that the minimal polynomial of $L'$ has smaller degree than that of $L$.

Consider a linear operator $L$ and some polynomial of it, $L'=p(L)$. Show that the minimal polynomial of $L'$ has degree less than or equal to the minimal polynomial of $L$. First, start working over ...
2
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0answers
91 views

Alternative proof that Harmonic sum is not an integer

In the post “Is there an elementary proof that ∑k=1/k is never an integer?” there is a simple and elegant proof by Bill Dubuque, who uses the prime 2 as a basis for his proof. I wondered whether a ...