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
46 views

Alternate proof for $a^2+b^2+c^2\le 9R^2$

As I studying geometric inequalities, one of those famous inequalities is $$a^2+b^2+c^2\le 9R^2$$ I did some research and I found that there is a proof (not exactly the this inequality but an useful ...
1
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0answers
15 views

Find similar estimate for complex numbers

According to Borwein, page 356 Prop. 2, $\left|\ln\left(\frac{4}{k}\right)-\operatorname{I}(1,k)\right|\leq 4k^2\left(8+\left|\ln k\right|\right)$ holds for $k\in\left(0,1\right]$. ...
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0answers
13 views

Conic sections and common functions

Is there a intuitive proof/reason of why plots of some common functions like y=x^2 are shaped like cross sections of a seemingly unrelated 3D object like a cone? ...
0
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0answers
40 views

Prove $f$ is Riemann integrable

Let $\{b_n\}$ be decreasing sequence converging to $0$. Define $f$ in terms of $b_n$ as follows $f(0)=1$ and $f(x)=0$ if $x$ is irrational, and $f(\frac{m}{n})=b_n$ if $x=\frac{m}{n}$ with ...
1
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0answers
25 views

Basic calculus question with continuous function [duplicate]

This is actually not my question, it was asked yesterday by user176744 in this link $[0,n]$ continuous function problem and I feel as if it didn't get enough attention. I am also interested in a ...
0
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1answer
53 views

Two reasons why $\int^{1}_{0}f(x) \,dx$ exists?

Consider $f$ on $[0,1]$ defined as $f(0)=0$ $$f(x)=2^{-n}\quad \text{if}\quad 2^{-n-1}<x\le2^{-n},$$ for $n=0,1,2,3,...$ I'm looking for two reasons why $\int^{1}_{0}f(x) \,dx$ exists? One ...
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1answer
32 views

how to prove the only difference between antidrivaties of a function is in their constants?

how to prove "If F is an antiderivative of f on an interval I , then the most general ...
7
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3answers
131 views

Inequality $\binom{2n}{n}\leq 4^n$

I would like to prove the following inequality, for $n=0,1,2,...$, $$ \binom{2n}{n}\leq 4^n.$$ I already proved it by induction, and I'm looking for another proof.
1
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2answers
43 views

Group of order $pq$ with $p\not\mid (q-1)$

Let $p, q$ be prime numbers, with $p<q$. If $G$ is a group of order $pq$ and $p\not\mid (q-1)$, then $G\cong \mathbb{Z}/pq\mathbb{Z}$. The standard way to prove this fact is using Sylow theorems, ...
4
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0answers
104 views

Will this algorithm stop before time?

For every $n \in \mathbb N$, let's define $a_0 = 0$, $$\begin{cases} a_{i+1} = 2a_i + 1 \pmod {2^n}, &\text{if it never appeared before} \\ a_{i+1} = 2a_i \pmod {2^n},& ...
0
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1answer
22 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 ...
0
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1answer
67 views

Prove that the graph $H = H_1\cup H_2 = (V_1\cup V_2,E_1\cup E_2)$ is connected.

Let $G = (V,E)$ be a graph and let $H_1 = (V_1,E_1)$ and $H_2 = (V_2,E_2)$ be two connected subgraphs of $G$ that have at least one node in common. Prove that the graph $H = H_1\cup H_2 = (V_1\cup ...
20
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1answer
396 views

Ask for alternative solution

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
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0answers
48 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
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2answers
68 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 ...
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0answers
48 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
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4answers
125 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
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1answer
27 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
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2answers
119 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
75 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
80 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
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0answers
29 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
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1answer
50 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
82 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
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0answers
89 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
111 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 ...
6
votes
2answers
119 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
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3answers
51 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
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2answers
59 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? $a_{n}=\frac{ln(n)}{n^2}$ --> $a_{n+1}=\frac{ln(n+1)}{(n+1)^2}$ $\frac{ln(n)}{n^2} > \frac{ln(n+1)}{(n+1)^2}$ ...
0
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0answers
18 views

Equivalence between the linear order of cardinal numbers and the axiom of choice

It is said here (paragraph History) that the Cantor-Bernstein theorem can be obtained easily from the linear order of cardinal numbers, and that the latter is equivalent to the axiom of choice. How ...
6
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2answers
101 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 ...
0
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0answers
39 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
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0answers
117 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
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1answer
54 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
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10answers
2k 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
219 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
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0answers
18 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
259 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
116 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 ...
1
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1answer
95 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
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2answers
95 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
92 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
57 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 ...
1
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1answer
74 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
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1answer
38 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
178 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
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1answer
39 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
76 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
61 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': ...
0
votes
2answers
67 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 ...