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
votes
1answer
508 views

Verify matrix identity $A^tD-C^tB=I$ on certain hypotheses

Given $n\times n$ real matrices $A,B,C,D$ such that: $AB^T$ and $CD^T$ are symmetric $AD^T-BC^T=I$ Prove that $A^TD-C^TB=I$ The solution I have come up with after a very long time is to consider: ...
3
votes
0answers
56 views

Is this irrationality proof correct?

Consider a non-square integer $n$. If its square root was rational, then we would have $$\sqrt n=\frac{a}{b}$$ for some $a,b\in\mathbb{Z}$ and so $a^2=nb^2$. But this is impossible, because $n$ is ...
3
votes
2answers
151 views

square root of 2 irrational - alternative proof

I have found the following alternative proof online. It looks amazingly elegant but I wonder if it is correct. I mean: should it not state that $(\sqrt{2}-1)\cdot k \in \mathbb{N}$ to be able to ...
1
vote
1answer
104 views

Measure of Elementary Sets Proof

I am struggling with what seems like a very simple problem from Terrence Tao's Introduction to Measure Theory book (which is available for free online by the way). What I am trying to prove is the ...
2
votes
4answers
279 views

Prove that the symmetric group $S_n$, $n \geq 3$, has trivial center.

I am trying to prove this: Let $\sigma$ be a non-identity element of $S_{n}$. If $n \geq 3$ show that $\exists \gamma \in S_{n}$ such that $\sigma\gamma \neq \gamma\sigma$. Hint: Let ...
0
votes
1answer
32 views

Simplified Galois proof?

I have learned about Galois epoch-making proof that any polynomial of the fifth degree has no solution representable in terms of its coefficients. Can his proof be simplified and clarified in modern ...
6
votes
2answers
209 views

Alternative proof of Girard's theorem

I am looking for an alternative proof of Girard's theorem. The standard proof, which is almost trivial, relies too much on visualizing spherical triangles on the sphere. Is there a more algebraic ...
2
votes
2answers
91 views

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $?

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $ for positive integer $n$ and real $a, b$? You can use any techniques you want. My proof just uses algebra, ...
3
votes
2answers
108 views

Characterize the natural numbers $n$ such that there is a surjective group homomorphism from $S_n$ to $S_{n-1}$.

This has always been one of my most favorite exercises from Group Theory, and I was surprised to see that this hasn't been asked before. To repeat: Characterize the natural numbers $n$ such that ...
0
votes
0answers
39 views

Proof that the $sqrt[k]{z}\, z \in \mathbb N$ counts the amount of numbers less than or equal to z with a $k-$exact power

Empirically, it can shown that $$\mathrm{Floor}[\sqrt[k]{z} ] \,, z \wedge k\in \mathbb N $$ is equal to the amount of numbers which have a $k-$exact root. For example, $\sqrt 36 = 6$ means that there ...
0
votes
1answer
196 views

Proving isomorphism between between a subspace and a quotient space

I've been thinking about this for a day or two now, and I think I've found a way to prove this, but am very unsure about how watertight this is: To be proven: Let $V$ be a vector space. If $V = U ...
3
votes
2answers
112 views

Prove withoui calculus: the integral of 1/x is logarithmic

It was known in the 17th century that the function $$ t \mapsto \int_{1}^{t} \frac{dx}{x} $$ is logarithmic: a geometric sequence in the domain produces an arithmetic sequence in the codomain. This ...
4
votes
0answers
96 views

Is it possible to find $\int \frac{1}{\sqrt[4]{1+x^4}} dx$ by parametrizing the curve $y^4-x^4=1$?

I found this integral in a handbook of integrals: $$\int \frac{1}{\sqrt[4]{1+x^4}} dx$$ I already have evaluated this integral by trigonometric substitutions and my answer agrees with that of the ...
0
votes
3answers
134 views

Prove $\lim\limits_{x\to 0^+} \frac{\ln x}{x} = -\infty$

Prove $\lim\limits_{x\to 0^+} \frac{\ln x}{x} = -\infty$ I've seen the following proof but I think it's invalid: $$\lim\limits_{x\to 0^+} \frac{\ln x}{x} = \lim\limits_{x\to 0^+}\ln x \cdot ...
7
votes
2answers
191 views

Having fun integral $\int_0^{\pi/4} \cos x \arctan(\cos x)\, dx$

Playing around with the inverse trigonometric function integration, I found a nice closed-form of the following integral $$\int_0^{\pi/4} \cos x \arctan(\cos x)\, ...
1
vote
3answers
62 views

Diagonalizability of a certain $4\times4$ matrix

Question $\bf 3.$ Determine if the following matrix is diagonalizable. (explain your answer) $$A=\pmatrix{ 1 & 4 & -2 & 3 \\ 3 & -3 & 0 & 4 \\ 1 & 1 & 1 ...
1
vote
2answers
70 views

How to prove that the sequence is decreasing $a_{n}=\frac{ln(n)}{n^2}$

Is my way/proof good and completely mathematically rigorous? $a_{n}=\frac{ln(n)}{n^2}$ --> $a_{n+1}=\frac{ln(n+1)}{(n+1)^2}$ $\frac{ln(n)}{n^2} > \frac{ln(n+1)}{(n+1)^2}$ ...
5
votes
2answers
148 views

Prove that $\mathbb{R}^{n}-A$ with the standard topology is connected where $n \geq 2$ and $A \subset \mathbb{R}^{n}$ is countable.

I've been stuck on this proof for quite a while. While I realize it is much easier to show using arcwise connectedness or pathwise connectedness, I would like to complete the proof without resorting ...
1
vote
0answers
76 views

Proof of “Every vector space has a basis $\implies$ AC” without mentioning von Neumann hierarchy

I am writing a short (30-50 pages) report on AC for an exam. I really would like to include the proof that "Every vector space has a basis $\implies$ AC". Actually, every proof I could find proves ...
0
votes
0answers
128 views

How to prove that a polynomial with leading coefficient $1$ has no fractional solutions

I want to prove that the equation $$x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots+a_0=0$$ has no solutions in the form of $p/q$ when $p$ and $q$ and coprime and $q>1$. With this polynomial, $a_n=1$ and ...
1
vote
1answer
74 views

Alternate proof for a theorem on ordered fields

I came across the following theorem, while studying "A First Course in Real Analysis" by Berberian Sterling. In an ordered field, if $a, b, c \geq 0$ and $a \leq b+c$, then $$ {{a}\over{1+a}} \leq ...
16
votes
10answers
3k views

Polygons with equal area and perimeter but different number of sides?

Let's say we have two polygons with different numbers of sides. They can be any sort of shape, but they have to have the same area, and perimeter. There could be such possibilities, but can someone ...
4
votes
3answers
480 views

Is there any theorem about figures of equal area and perimeter being congruent?

I had an idea, that all geometric objects, that are different, as they're not a translation, rotation, and a reflection of one another cannot have the same area AND perimeter, as compared to ONE ...
0
votes
0answers
23 views

Business Stats Question Please Help

For students taking online courses at ABC University, grades have been distributed as follows: A – 40%, B – 30%, C – 15%, D – 10% and F – 5%. A sample of 173 students taking online courses at KSU has ...
5
votes
5answers
297 views

Elementary theorems with several proofs?

Every year my student's math club organizes a "proof marathon", where we present multiple proofs for a single theorem. For instance, last edition we did the AM-GM inequality with geometric, algebraic, ...
4
votes
1answer
258 views

How to prove Post's Theorem by induction?

The proof of post's theorem is given in my textbook in two pages of explanation using a non-induction method. I was told that ,using induction on length of the proof, one can get a simpler and more ...
2
votes
1answer
106 views

Check proof please

Prove that if: $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ than: $\lim_{x\rightarrow\infty}{\frac{f(x)}{x}}=L$ Assuming $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ we can choose $X_{\epsilon}$ s.t. ...
4
votes
2answers
163 views

Proof about AM-GM inequality generalized

Note: I'm not sure this type of questions are welcome on the site. In case tell me. Let's define the $p$ mean as $$M_p(x_1, \dots, x_n) = \sqrt[p] { \frac 1n \sum_{i = 1}^n x_i^p}$$ for $x_1, ...
2
votes
3answers
98 views

If $I_{n+1}\subset I_n$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty

Question: If $I_n$ is closed and bounded, $I_{n+1}\subset I_n$, and $I_n\neq\emptyset$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty. This is not a homework help question. I'm actually looking ...
5
votes
1answer
84 views

All $k$-regular subgraphs of $K_{n,n}$ have a perfect matching: a proof without Hall's Marriage Theorem?

There are several ways of describing this result: Theorem: For $k \in \{1,2,\ldots,n\}$, any $k$-regular subgraph of $K_{n,n}$ has a perfect matching (also known as a $1$-factor). I tend to ...
2
votes
1answer
93 views

Ratio of 2 Gammas, approximation with power

Find all value of $\alpha$ such that $\lim\limits_{x\rightarrow +\infty}\left(\frac{\Gamma(x+\alpha)}{\Gamma(x)}-x^{\alpha}\right)=0$. (note: $\alpha$ is a constant with respect to $x$) By ...
3
votes
1answer
46 views

The number of holomorphic coverings (with given degree) of the punctured sphere is finite.

I'm looking for a proof of the following theorem: Fix a finite set $B=\{y_1,\ldots,y_k\}\subseteq \mathbb P^1(\mathbb C)$, then there is only a finite number of isomoprhism classes of ...
2
votes
5answers
517 views

$ABC=I\implies B$ is invertible and $B^{-1} = CA$

$A$, $B$ and $C$ are square matrices with $ABC=I$. I need to show that $B$ is invertible and $B^{-1} = CA$. I have proved it using the fact stated here. Since we only need to prove invertiblity of ...
0
votes
1answer
42 views

Different approaches for this problem (number in different bases)

I came across the following problem on a puzzle website the other day. A $3$ digit number is written $(xyz)_{10}$ in base $10$ and $(zyx)_9$ in base $9$. What is it? The website (which I think ...
2
votes
3answers
99 views

$x^{1+\epsilon}$ is not uniformly continuous on $[0,\infty)$

There are two questions. First: is the proof underneath correct? Let $\epsilon>0$ and let $f(x)=x^{1+\epsilon}$. I aim to show that $f$ is not uniformly continuous on $[0,\infty)$. We will show ...
2
votes
2answers
79 views

Epic-monic factorisation in $\mathbf{Set}$.

I'm stuck on Exercise 5.2.1 of Goldblatt's "Topoi: A Categorial Analysis of Logic": Given a function $f:A\to B$, if $h\circ g: A\twoheadrightarrow C\rightarrowtail B$ and $h'\circ g': ...
4
votes
0answers
88 views

${n \choose r}=\frac {n!}{r!(n-r)!}$ without using the permutation approach.

I had an idea that would be to first prove Pascal's Rule, $${n \choose r} = {n-1 \choose r-1} + {n-1 \choose r},$$ which can be proved combinatorically whether one particular element (among the $n$) ...
0
votes
2answers
89 views

Proof that doesn't exists a rational $s$ such that $s^2 = 6$

Well, I solved it, and I would like to know if there is anything that can be corrected or improved here. I think that the proof ended up too long, and with too many letters. Surely there is a better ...
2
votes
1answer
28 views

Is this alternate answer right?

This is really basic, but I don't want to get the fundamentals wrong. Show that if $A$ is closed in $Y$ and $Y$ is closed in $X$, then $A$ is closed in $X$. The solution uses subspaces (which I ...
2
votes
3answers
29 views

Help complete proof ('or' statement in conclusion): $a^2 \ge 7a \Rightarrow a\le0 \text { or } a\ge7$

Question Prove that if $a$ is a real number such that $a^2 \ge 7a$ then $a\le0 \text { or } a\ge7$ My Attempt We are given: $a \in \mathbb{R}$ $a^2 \ge 7a$ And need to prove: $a\le 0 \text{ ...
0
votes
1answer
50 views

Give alternative proof of given theorem (relates to inverses of functions) need to use (b,a)∈ $B\times A.$

I have no idea of how to do this. We need to create an alternative proof using some of the ideas on the bottom, but I'm lost. Any ideas on how to do this? I'm not sure how to even start the ...
6
votes
1answer
225 views

Area of ​​the intersection of two discs: Integral solution?

Here is the problem : We consider two circles that intersect in exactly two points. There $O_1$ the center of the first and $r_1$ its radius. There $O_2$ the center of the second and $r_2$ its ...
4
votes
7answers
259 views

Alternative Proof of Infinitely Many Primes? [duplicate]

I've seen Euclid's proof of infinitely many primes, what are other approaches to proving there are infinitely many primes?
1
vote
1answer
58 views

Prove that $\dfrac{\sigma_1(n)}{n} = \sigma_{-1}(n)$ where $\sigma_x(n)$ is the sum of the $x$th powers of the positive divisors of $n$.

I computed $\dfrac{\sigma_1(n)}{n}$ and $\sigma_{-1}(n)$ on a good hundred values of $n$, and they seem to always match. For example: $\dfrac{\sigma_1(6)}{6} = \dfrac{1 + 2 + 3 + 6}{6} = ...
1
vote
2answers
102 views

Prove that n!+1 contains a prime factor greater than n and use this to prove that there are infinte many primes

Prove that $n!+1$ contains a prime factor greater than $n$ and use this to prove that there are infinitely many primes. I said assume that $n!+1$ contains a prime $p$ which is less than or equal to ...
0
votes
1answer
48 views

Convergence of Alternating harmonic series (Direct!)

Once again, note No use of the alternating series test!
0
votes
1answer
49 views

Left & right adjoints in the context of complete lattices.

This is a follow-up question from this question of mine. In the same paper as the one mentioned in my previous post, it's stated that In the context of complete lattices, a monotone map has a ...
4
votes
2answers
119 views

inequality $\prod\limits_{k=1}^n\frac{2k-1}{2k}\lt\frac1{\sqrt{3n}}$

$n$ is a positive integer, then $$\prod_{k=1}^n\frac{2k-1}{2k}\lt\frac1{\sqrt{3n}}$$ with mathematical induction, we can prove this. But I would love to find a wonderful method without ...
1
vote
1answer
54 views

Orbit-Stabiliser Theorem Application

Question Let $G$ be the symmetric group $S_n$ acting on the $n$ points $\lbrace 1, 2, 3, . . . , n\rbrace$, let $g\in S_n$ be the n-cycle $(1,2,3,. . . , n)$. By applying the Orbit-Stabiliser Theorem ...
11
votes
1answer
141 views

Seeking a more direct proof for: $m+n\mid f(m)+f(n)\implies m-n\mid f(m)-f(n)$

If $f:\mathbb N\to\mathbb Z$ satisfies: $$\forall n,m\in\mathbb N\,, n+m\mid f(n)+f(m)$$ How to show that this implies: $$\forall n,m\in\mathbb N,\,n-m\mid f(n)-f(m)?$$ I was almost incidentally ...