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

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21 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
23 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 ...
2
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4answers
92 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
26 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
53 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
22 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 ...
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0answers
55 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 ...
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8answers
330 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 ...
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2answers
59 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 ...
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0answers
45 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 ...
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2answers
86 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
53 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
16 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
67 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
45 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
26 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 ...
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1answer
128 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
54 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 ...
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1answer
27 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 ...
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1answer
27 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 ...
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0answers
87 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 ...
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50 views

$g$ o $f$ is nowhere continuous

Find an example of a function $f$ discontinuous on $\mathbb{Q}$ and another function $g$ discontinuos at only one point, but $g$ o $f$ is nowhere continuous. One solution is: Let $f$ be the ...
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0answers
24 views

Solutions to the heat equation given certain growth conditions

Let $u(x, t)$ be the solution of the following Cauchy problem for the heat equation given by \begin{align*} u_{t}(x, t) - u_{xx}(x, t) &= 0 \quad (x, t) \in \mathbb{R} \times (0, T)\\ u(x, 0) ...
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0answers
40 views

coloring theorem for topological partitions

Let $(X,\tau)$ be a topological space. DEFINITIONS: Define a topological partition of $X$ into connected sets to be a collection of pairwise disjoint open connected sets $\{U_i\}_{i\in I}$ such that ...
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1answer
51 views

Prove the operator is positive

I'm searching for an alternative proof of the following: Let $U$ be a self-adjoint operator on a Hilbert space $H$, define $m=\inf_{\|x\|=1}\langle Ux,x\rangle$ and $M=\sup_{\|x\|=1}\langle ...
4
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3answers
139 views

Proof of no primes such that $x^2 + y^2 = z^2$ [duplicate]

I'm in a pretty simple "CS Math" course for year 1 Comp Sci, and I came across this: Disprove, $x^2 + y^2 = z^2$, such that $x, y, z$ are primes I thought of this as, if n is a prime, then prime ...
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1answer
154 views

Stefan-Boltzmann Constant and Stefan's Law

The following argument is from my textbook, An Introduction to Thermal Physics by Daniel Schroeder. If you are familiar with the derivation of Stefan's Law from the energy density of a photon gas, you ...
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2answers
41 views

How to prove this simple fact without using distribution theory?

Suppose function $f(x) $ is normalized to unity, i.e., $$ \int dx |f(x)|^2 =1 . $$ Now consider the Fourier transform of $f$, i.e., $$ F(k) = \int d x f(x) e^{-i k x} . $$ Here we assume that $f $ ...
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0answers
30 views

Proof: The reduced row echelon form of a matrix is unique.

If $A \in M_{m\times n}$ with real entries, then there exist a unique matrix $R$ in row echelon form such that $A\sim R$, where $R$ comes from $A$ after performing elementary operations. How can I do ...
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10answers
144 views

How to prove $x^3-y^3 = (x-y)(x^2+xy+y^2)$ without expand the right side?

I can prove that $x^3-y^3 = (x-y)(x^2+xy+y^2)$ by expanding the right side. $x^3-y^3 = (x-y)x^2 + (x-y)(xy) + (x-y)y^2$ $\implies x^3 - x^2y + x^2y -xy^2 + xy^2 - y^3$ $\implies x^3 - y^3$ I was ...
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2answers
112 views

$X$ compact Hausdorff space, characterize the maximal ideals of $C(X)$

I know this question has been asked before, but I think I'm very close to a new solution and wanted to know if it is a viable approach. Let $C(X)$ be the ring of continuous functions $X \rightarrow ...
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0answers
49 views

squeeze theorem - math

I am trying to prove the following: 1/ek <= (1/k)(1-(1/k))^(k-1) <= 1/2k for k>=2 in doing so I tried induction proof, and contradiction and it didn't work, it gets too complicated... Then ...
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1answer
138 views

Proving the Sine Rule with one line.

Working on a general proof of the Law of Sines for ALL Euclidean triangles. Right triangles are easy. Acute triangles are just two proofs of the right triangle. But this is not sufficient for me. I ...
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3answers
113 views

Show that $\frac{1}{a}+\frac{1}{b}\not=\frac{1}{a+b}$

Problem Assume that $a,b\in\mathbb{R}-\{0\}$ and that $a+b\not=0$. Prove that $\frac{1}{a}+\frac{1}{b}\not=\frac{1}{a+b}$. My Proof Let's assume that $\frac{1}{a}+\frac{1}{b}=\frac{1}{a+b}$, then ...
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2answers
31 views

Concise proof of $\operatorname{dim} V =\operatorname{dim\,ker}T+\operatorname{dim \, range}T$ [duplicate]

Can I have a concise proof of the following: $$\operatorname{dim} V =\operatorname{dim\,ker}T+\operatorname{dim \, range}T$$ I have read a few proofs of this, and they are all so long, I always ...
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1answer
26 views

Definition of Ordinal (w/ Axiom of Regularity) (problem 37, page 208, Enderton's Elements of Set Theory)

Given the definition of an ordinal to be well-ordered by $\in$ and transitive, I am interested with proving the following: I know and understand the following which easily proves it, but uses the ...
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1answer
39 views

A possible alternate proof for linear independence

I am wondering about the theorem If $T: V \to U$ is a linear non-singular transformation and $\{v_1,..,v_k\}$ is a linearly independent subset of V then the images of T are also independent. I know ...
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1answer
39 views

Alternative proof for a probability question

There is a probability space $(\Omega,\mathcal A, P)$ and random variables $X:\Omega \to \mathbb R$ and $Y:\Omega \to \mathbb R$, show that: $\{\omega | X(w) \le Y(w)\} = \{ X \le Y \} \in \mathcal ...
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0answers
47 views

Looking for an alternative solution for the mutilated chessboard problem

Given a mutilated chessboard where two diagonally opposite squares are missing (the unmutilated version of it has $64$ squares), and given $31$ domino pieces, is it possible to cover the entire ...
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0answers
51 views

A Theorem On Compact Connected Metric Spaces by Stadje

I recently came across a surprising theorem, due to Wolfgang Stadje, a special case of which states that: Let $(X,d)$ be a compact connected metric space. Then there exists a unique real number ...
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3answers
48 views

Understanding proof that rationals of the form $\frac{p}{2^n}$ are dense in $\mathbb{R}$

Only the following definitions may be used in the proof: 1) A set $X$ is dense in $\mathbb{R}$ if the closure of $X = \mathbb{R}$. 2) The closure of $X$ is $X \cup X'$, where $X'$ is the ...
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4answers
54 views

Prove that if the identity is written as the product of $r$ transpositions, then $r$ is an even number

Theorem. If the identity is written as the product of $r$ transpositions, $id=τ_1τ_2\dots τ_r$, then $r$ is an even number. Proof. We will employ induction on $r$. A transposition cannot be the ...
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1answer
75 views

Prove that there exists irrational numbers $x$ and $y$ such that $x + y$ is rational, without using subtraction

My homework has this problem: Prove that there exist irrational numbers $x$ and $y$ such that $x + y$ is rational. There is an easy solution that I found on mathbitsnotebook.com: ...
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62 views

Is information entropy $H(X)$ a sub modular function?

I was trying to learn more about sub modular functions and wanted to see an example of proving that some function is sub modular. Wikipedia said that Entropy was an example so I decided to try it out ...
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1answer
45 views

Is this proof by counterexample valid?

This is the given statement and its proof: $$\exists m \in Z^+, \forall n \in Z^+, m<n$$ Proof: This result is false because, for each positive integer m, if we put $n=m$ then n is a positive ...
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1answer
33 views

Measure theoretic proof of $|\Bbb{Z}^d/A\Bbb{Z}^d| = |\det(A)|$

Let $A \in \Bbb{Z}^{d\times d}$ be an invertible matrix with entries in $\Bbb{Z}$. It is well-known (and can be proved using algebraic properties of matrices) that the index of the group $A \Bbb{Z}^d ...
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0answers
29 views

Prove without method of contradiction that there exists a real number less than every positive real number that is positive

This question was asked before for proof by contradiction and which got me into thinking whether i could prove it without using a contradiction Original problem statement is here Prove by ...
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1answer
43 views

Proving continuity of $f(x)=x\cos(2\pi/x)$ at $x=0$

I know that the function $f(x)=x\cos(2\pi/x)$ if $x\neq0$ and $f(0)=0$ is continuous at $x=0$ using $\epsilon-\delta$ as follows: $\lvert x\cos(2\pi/x)\rvert=\lvert ...
2
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1answer
46 views

An inequality with elementary symmetric polynomials

Fix a natural number $n\geq 1$. Let $a_1, \ldots, a_n$ be $n$ real numbers such that $a_i>0$ for each $i$. Show that for each natural $k$ with $0\leq k\leq n$ $$e_k(a_1,\ldots, ...