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

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2answers
24 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 ...
5
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1answer
87 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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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 ...
2
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2answers
133 views

Proof for $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$ without complexes? [duplicate]

This is what I needed. Practically, a link were also okay. $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$
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1answer
298 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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0answers
21 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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4answers
4k views

Why Markov matrices always have 1 as an eigenvalue

Also called stochastic matrix. Let $A=[a_{ij}]$ - matrix over $\mathbb{R}$ $0\le a_{ij} \le 1 \forall i,j$ $\sum_{j}a_{ij}=1 \forall i$ i.e the sum along each column of $A$ is 1. I ...
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1answer
474 views

Problem with alternate solution — Equation of plane through point and containing intersection line of two planes [Stewart P $803, 12.5.37$]

$37.$ Find an equation of the plane that passes through the point $(1, -2, 1)$ and contains the line of intersection of the planes $x + y - z = 2$ and $2x - y + 3z = 1$. $\bbox[3px,border:2px solid ...
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0answers
22 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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0answers
13 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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0answers
52 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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6answers
667 views

A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$

This is a problem from "A Course of Pure Mathematics" by G H Hardy. Find the limit $$\lim_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$$ I had solved it long back (solution presented in my blog ...
2
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0answers
60 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 ...
3
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2answers
70 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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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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0answers
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 ...
3
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0answers
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 ...
0
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1answer
759 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 ...
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0answers
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 ...
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1answer
39 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 ...
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2answers
84 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 ...
2
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2answers
71 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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0answers
69 views

Alternative Proof for “Roots of Mertens Function-Farey Sequence-Cosines Relations”

You can write Merten's function as $$ M(n)= \sum_{a\in \mathcal{F}_n} e^{2\pi i a} , $$ where $\mathcal{F}_n$ is the Farey sequence of order $n$. The sum may be split into imaginary and real ...
4
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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 $\mu, \nu, \mu_1, \mu_2, \ldots$ ...
5
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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 ...
15
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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 ...
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 ...
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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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
46 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 ...
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2answers
50 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 ...
0
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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 ...
1
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1answer
96 views

Cutting chocolate diagonally

Given is chocolate with rectangular pieces of size $a \times b$. If cut diagonally, how many pieces will it be split into? If knife passes exactly by co-catenating we assume there is no damage to ...
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0answers
45 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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0answers
220 views

Is independence a transitive property? [duplicate]

If the events $A$ and $B$ are independent and the events $B$ and $C$ are independent, does this necessarily mean events $A$ and $C$ are independent? I used coin tosses to try to model this with $A = ...
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1answer
51 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 ...
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 ...
4
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3answers
310 views

Prove that $\sqrt[3]{5-\sqrt{2}}$ is not a rational number

Prove that $\sqrt[3]{5-\sqrt{2}}$ is not a rational number My attempt: Consider the polynomial $ (x^3-5)^2 - 2 = x^6 -10x^3 + 23 = 0 $. By the rational root theorem, we can conclude that $ \pm ...
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1answer
221 views

$\log_2 13$ is irrational

Is it true that $\log_2 13$ is irrational? Let $x=\log_2 13\implies 2^x=13$. So, it will be an irrational number, if not,$$x=\frac p q$$ and $$2^{\frac p q}=13$$ $$\implies 2^p=13^{q}$$ Since, ...
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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 ...
16
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3answers
1k views

Prove that $x^3 + y^3 = z^3$ has no integer solutions as briefly as possible

Can someone provide the proof of the special case of Fermat's Last Theorem for $n=3$, i.e., that $$ x^3 + y^3 = z^3, $$ has no positive integer solutions, as briefly as possible? I have seen some ...
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
62 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 ...