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

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2
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2answers
53 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 ...
0
votes
2answers
29 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 ...
2
votes
2answers
45 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 ...
1
vote
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 ...
3
votes
1answer
124 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
votes
1answer
47 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
votes
1answer
20 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
votes
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 ...
2
votes
0answers
86 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 ...
0
votes
0answers
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
23 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
36 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 ...
3
votes
1answer
47 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
votes
3answers
132 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 ...
7
votes
1answer
139 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 ...
5
votes
2answers
40 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 $ ...
0
votes
0answers
27 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 ...
2
votes
10answers
141 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 ...
2
votes
2answers
106 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 ...
0
votes
0answers
46 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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vote
1answer
136 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 ...
7
votes
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 ...
0
votes
2answers
30 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 ...
0
votes
1answer
24 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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
0answers
43 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 ...
3
votes
0answers
50 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 ...
1
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3answers
46 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 ...
6
votes
4answers
47 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 ...
2
votes
1answer
72 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: ...
0
votes
0answers
46 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 ...
1
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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 ...
3
votes
1answer
32 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 ...
1
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0answers
27 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 ...
1
vote
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
votes
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, ...
1
vote
0answers
73 views

Partition of unity of Lagrange polynomials

Given a sequence of increasing real numbers $T = \{t_1 < t_2 < ... < t_{d+1}\}$, the $d+1$ Lagrange polynomials $L_i(t)$ of degree $d$ are defined as $$L_i(t) = \prod_{\substack{1\leqslant j ...
3
votes
1answer
21 views

Other proof for existence of monotone subsequences

Is there any other proof of Bolzano-Weierstrass theorem (i.e.: Let ${\{x_n}\}$ be an arbitrary sequence of real numbers. Then ${\{x_n}\}$ has a monotone subsequence.), WITHOUT using concept of ...
1
vote
2answers
65 views

Alternative proof of a transpose property

I am asked to prove; $$(AB)^T=B^TA^T$$ although it is very simple to prove it by the straight forward way, in the exercise I am asked to prove it without using subscripts and sums, directly from the ...
2
votes
2answers
69 views

Is there another way to prove $(x-n)^2 = (n-x)^2$

Let's say $n$ is $4$. So, I came up with the solution below. $(x-4)^2 = (x-4)(x-4) = x^2 - 8x + 16$ $(4-x)^2 = (4-x)(4-x) = 16 - 8x + x^2 = x^2 - 8x + 16$ I was wondering if there is another way ...
0
votes
1answer
67 views

Proving Submultiplicativity on a Matrix Norm

Let $||A||=(\sum_{i=1}^{n}\sum_{j=1}^{n}{a_{ij}^p})^{1/p}$, and let p=2. Then prove that $\|AB\|\le \|A\|\|B\|$ I have looked at numerous proofs for this, and I don't see one that satisfies me ...
2
votes
0answers
50 views

Is it possible to prove that $\text{tr}(AB)=\text{tr}(BA)$ without using matrices? [duplicate]

Is it possible to prove that $\text{tr}(AB)=\text{tr}(BA)$ without using matrices or having to choose a particular base ? Such a proof should probably use a non matricial definition of traces. One ...
1
vote
2answers
59 views

How to get to $5^3 \geq n^3$ in the proof by contradiction?

This is the same problem asked here. - Next step to take to reach the contradiction? Here is it again. I understand the solution - how you want to get to the fact 100 divides n^2 and then go ...
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vote
1answer
38 views

e Online source for alternative proofs

I'm looking for some alternative proofs for various theorems. My goal is to compile a list of various proofs each relating to a specific theorem (such as the triangle inequality, Fermat's Little ...
4
votes
6answers
162 views

Prove that $x-1$ is a factor of $x^n-1$

Prove that $x-1$ is a factor of $x^n-1$. My problem: I already proved it by factor theorem† and by simply dividing them. I need another approach to prove it. Is there any other third ...
11
votes
4answers
203 views

Checking that a $3$-D diagram is commutative

When proving certain results I need to use commutative diagrams, some of which quite complicated. My question is: Do we need to check every small square all the time to make sure that they are all ...
15
votes
6answers
566 views

Fascinating induction problem with numerous interpretations

Problem: Suppose you begin with a pile of $n$ stones and split this pile into $n$ piles of one stone each by successively splitting a pile of stones into two smaller piles. Each time you split a pile, ...
0
votes
2answers
47 views

Trouble solving this induction problem

Show that, for every $n\ge2$, $3^n >n(n-1)$. Well, I started by showing the base case ($n = 2$): $3^2 > 2$ Now, for $n+1$: $P(n)\Rightarrow P(n+1)$ $$3^{n+1} > (n+1)n$$ My ...
1
vote
1answer
52 views

Can someone verify my proof by contraposition?

This is a problem from Discrete Mathematics and its Applications Is there a way to tell right away what type of proof to use or does that just come with practice (build intuition - oh here i ...