For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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6
votes
1answer
2k views

Can you please check my Cesaro means proof

I wanted to prove the following: if $x_n \to x$ then $y_n \to x$ where $$ y_n = {x_1 + \dots + x_n \over n}$$ Please can you tell me if my proof is correct? My proof is this: Let $\varepsilon > ...
24
votes
5answers
4k views

Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$

I've just started to learn about the tensor product and I want to show: $$(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}.$$ Can you tell ...
25
votes
10answers
4k views

Proving $\sqrt 3$ is irrational.

There is a very simple proof by means of divisibility that $\sqrt 2$ is irrational. I have to prove that $\sqrt 3$ is irrational too, as a homework. I have done it as follows, ad absurdum: Suppose ...
8
votes
2answers
2k views

Nested Radical of Ramanujan

I think I have sort of a proof of the following nested radical expression due to Ramanujan for $x\ge 0$. $$\large x+1=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+\cdots}}}}$$ for $ x\ge -1$ I ...
5
votes
2answers
137 views

Analytic $F(z)$ has $f(z)$ as derivative $\implies$ $\int_\gamma f(z)\ dz = 0$ for $\gamma$ a closed curve

Hypothesis: Suppose that $F(z)$ has $f(z)$ as a derivative. Suppose further that $F(z)$ is analytic. Now consider the complex line integral $$ \tag{1} \int_\gamma f(z)\ dz $$ Question: Does this ...
14
votes
3answers
2k views

No simple group of order $300$

So I've been trying to prove that there's no simple group of order $300$. This is what I did and I was wondering if it was enough. $|G|=2^2 \cdot 3 \cdot 5^2$. Suppose $G$ is simple. Then there ...
3
votes
3answers
659 views

Is this proof of the fundamental theorem of calculus correct?

A student friend of mine recently gave me a proof of the fundamental theorem of calculus which does not correspond to any I can find in the textbooks. It starts by considering an increasing continuous ...
0
votes
2answers
69 views

Proof of certain Gaussian integral form

I am having trouble understanding where the following integral form comes from: $$\int_{-\infty}^{\infty} e^{-a x^2 }e^{-bx}=\sqrt{\frac{\pi}{a}}e^{\frac{b^2}{a}}$$ I see and understand that the value ...
1
vote
1answer
266 views

How to prove Greatest Common Divisor using Bézout's Lemma

The problem is to prove the following If $\gcd(a,b) = c$, then $\gcd(a^m, b^m) = c^m$ I know that this can be solved easily by proving that $c\mid a \implies c^m \mid a^m$ and $c\mid b \implies ...
2
votes
1answer
217 views

every Abelian group is a converse lagrange theorem group

Let $G$ be a finite abelian group, then $G$ has a subgroup of order $n$ if and only if $n\mid G$. Proof: by Lagrange if $H\leq G$ then $|H|$ divides $|G|$ so this proves one of the implications. We ...
3
votes
3answers
637 views

Prove $ (A \cup B) \cap C$ = $(A \cap C) \cup (B \cap C) $

Prove $ (A \cup B) \cap C$ = $(A \cap C) \cup (B \cap C) $ Starting from the left side, $ (A \cup B) \cap C = $ By distributive law, ( distributing the $\cap C$), we have $ (A \cap C ) \cup (B ...
1
vote
4answers
106 views

If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$ [duplicate]

If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$ So far I've got: $a<b$ $a^2<ba$ $a<\sqrt{ab}$ And: $a<b$ $a+b<2b$ $\frac{a+b}{2}<b$ So I need to prove ...
0
votes
1answer
3k views

Prove or disprove - If a divides b and b divides a does a=b

Prove or disprove: If a, b belong to the set of positive integers, and if a divides b and b divides a, then a=b. Does this hold if if a,b are not necessarily positive? Why or Why not? Here is what I ...
4
votes
1answer
5k views

Need verification - Prove a Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues

Proof: Let eigenvalue $\lambda \neq \vec{0}$ such as $$\textbf{A}\vec{v} = \lambda\vec{v}$$ $$\Rightarrow (\textbf{A}\vec{v})^\ast = (\lambda\vec{v})^\ast$$ $$\Rightarrow ...
2
votes
5answers
413 views

Prove that $\sin^2(A) - \sin^2(B) = \sin(A + B)\sin(A -B)$

This is my attempt: $$ \begin{align} & \phantom{={}}\sin^2(A) - \sin^2(B) = \sin(A + B)\sin(A -B) \\[8pt] & = (\sin(A)\cos(B)+\cos(A)\sin(B))(\sin(A)\cos(B) - \cos(A)\sin(B)) \\[8pt] & = ...
15
votes
4answers
3k views

Proving that there are infinitely many primes with remainder of 2 when divided by 3

I need to prove that there are infinitely many primes with remainder of 2 when divided by 3. I started out similarly to Euclid's classic proof of an infinite number of prime numbers: Suppose there is ...
10
votes
6answers
2k views

Simpler proof - Non atomic measures

Suppose that $(X,\mathcal{E},\mu)$ is a non-atomic finite measure space (i.e. for every $E \in \mathcal{E}$ with $\mu(E)>0$ there exists $F \subset E$ measurable such that $0<\mu(F) ...
8
votes
3answers
184 views

Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.

Problem: Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$. My work: So I think I have to do a proof by induction and I just wanted some help editing my proof. My ...
4
votes
2answers
2k views

Proof Verification - Every sequence in $\Bbb R$ contains a monotone sub-sequence

Came across the following exercise in Bartle's Elements of Real Analysis. This is the solution I came up with. Would be grateful if someone could verify it for me and maybe suggest better/alternate ...
5
votes
1answer
2k views

Continuous images of compact sets are compact

Let $X$ be a compact metric space and $Y$ any metric space. If $f:X \to Y$ is continuous, then $f(X)$ is compact (that is, continuous functions carry compact sets into compact sets). Proof: ...
3
votes
2answers
25k views

Proof by induction: $2^n > n^2$ for all integer $n$ greater than $4$ [duplicate]

I am a CS undergrad and I'm studying for the finals in college and I saw this question in an exercise list: Prove, using mathematical induction, that $2^n > n^2$ for all integer n greater than ...
2
votes
3answers
213 views

de morgan law $A\setminus (B \cap C) = (A\setminus B) \cup (A\setminus C) $

First part : I want to prove the following De Morgan's law : ref.(dfeuer) $A\setminus (B \cap C) = (A\setminus B) \cup (A\setminus C) $ Second part: Prove that $(A\setminus B) \cup (A\setminus ...
1
vote
3answers
1k views

Proof of Drinker paradox [duplicate]

I searched all over the internet but didn't find a formal proof for this paradox, so here is my attempt: $\exists x[P(x)\implies \forall yP(y)]$ Let $x=x_0$. Thus $P(x_0)$ is given. Let $y$ be ...
1
vote
0answers
65 views

Proof by contradiction: $E_1+E_2\doteq E_1 \oplus E_2 \leftrightarrow E_1 \cap E_2=\{0_V\}$

I must proof the following: Prop.: Let $E_1,E_2$ two vector subspace of $V$ then $$E_1+E_2\doteq E_1 \oplus E_2 \leftrightarrow E_1 \cap E_2=\{0_V\}$$ Proof: I must show $$1)E_1+E_2\doteq E_1 \oplus ...
4
votes
4answers
2k views

If $f,g$ are uniformly continuous prove $f+g$ is uniformly continuous

Suppose $f:E \rightarrow \mathbb{R}$ and $g:E \rightarrow \mathbb{R}$ are uniformly continuous, where $E$ is a subset of $\mathbb{R}$. Show that $f+g$ is uniformly continuous. What about $fg$ and ...
4
votes
2answers
1k views

Supremum and infimum of $\{\frac{1}{n}-\frac{1}{m}:m, n \in \mathbb{N}\}$

I would like to verify my proof of the following: Let $A=\{\frac{1}{n}-\frac{1}{m}:m, n \in \mathbb{N}\}$. I want to show that $-1$ and $1$ are the infimum and supremum respectively. First I will ...
2
votes
2answers
253 views

basic induction probs

Hello guys I have this problem which has been really bugging me. And it goes as follows: Using induction, we want to prove that all human beings have the same hair colour. Let S(n) be the ...
45
votes
12answers
7k views

Proof that every number ≥ $8$ can be represented by a sum of fives and threes.

Can you check if my proof is right? Theorem. $\forall x\geq8, x$ can be represented by $5a + 3b$ where $a,b \in \mathbb{N}$. Base case(s): $x=8 = 3\cdot1 + 5\cdot1 \quad \checkmark\\ x=9 = 3\cdot3 ...
17
votes
4answers
804 views

Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$

For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$. My attempt: Let $$\begin{align*} f_n(x) &= \frac{\ln\left(1-2 ...
8
votes
3answers
524 views

$f^*dx_i = \sum_{j=1}^l \frac{\partial f_i}{\partial y_j} dy_j = df_i$

Guillemin and Pollack's Differential Topology Page 164: $U \subset \mathbb{R}^k$ and $V \subset \mathbb{R}^l$ be open subsets. Let $f: V \to U$ to smooth. Use $x_1, \dots, x_k$ for the standard ...
2
votes
1answer
642 views

Normal subgroups of $S_n$ for $n\geq 5$.

Question is to : Find all normal subgroups of $S_n$ for $n\geq 5$. What I have done so far is : We know that $A_n$ is one normal subgroup of $S_n$. Suppose $H\neq (1)$ is another normal subgroup ...
15
votes
2answers
1k views

Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more ...
8
votes
2answers
139 views

Ways to prove Eulers formula for $\zeta(2n)$

I recently, out of interest, tried to prove Euler's formula $\zeta{(2n)}=(-1)^{n-1}\frac{(2\pi)^{2n}}{2(2n)!}B_{2n}$ for all $n\in\mathbb{N}$. I adapted Euler's original proof for ...
5
votes
1answer
4k views

Irreducible but not prime in $\mathbb{Z}[\sqrt{-5}] $

Show that $2,3, 1-\sqrt{-5}, 1+\sqrt{-5}$ are irreducible over $\mathbb{Z}[\sqrt{-5}]$, but not prime and that 1 and -1 are the only units. Let $N$ be the norm map into $\mathbb{Z}$ and let u ...
4
votes
1answer
1k views

If a subsequence of a Cauchy sequence converges, then the whole sequence converges.

Let $(X,d)$ be a metric space, and say $(x_n)$ is a Cauchy sequence such that it has a convergent subsequence $(x_{n_k})$ that converges to $x$. We show $x_n \to x$. Let $\epsilon > 0$. Take $N ...
2
votes
1answer
653 views

Show that the Kelvin-transform is harmonic

First, I have to give you our definitions: Consider $\Omega:=\mathbb{R}^n\setminus\overline{B}_R(0)$ with $R>0$ and $n>1$. The function $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ ...
7
votes
2answers
469 views

If $\gcd(m,n)=1$, then $\mathbb{Z}_n \times \mathbb{Z}_m$ is cyclic. [duplicate]

If $\gcd(m,n)=1$, then $\mathbb{Z}_n \times \mathbb{Z}_m$ is a cyclic group. Let's denote $\mathbb{Z}_n=\langle1_n \rangle$ and $\mathbb{Z}_m=\langle1_m \rangle.$ My proof goes as follows: since ...
0
votes
1answer
200 views

Prove Z is a martingale by defining it is a product of random variables

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}, \mathbb{P})$ where $\mathscr{F_n} = \mathscr{F_n}^{Z} \doteq \sigma(Z_0, Z_1, \ldots, Z_n)$, show that $Z = (Z_n)_{n \geq ...
7
votes
3answers
2k views

Show that G is a group, if G is finite, the operation is associative, and cancellation law hols

Let $G$ be a non-empty finite set with an associative binary operation so that cancellation law holds, i.e. $ab=ac$ or $ba=ca$ implies $b=c$, for any choices of $a,b,c$ in $G$. Assume that there is an ...
3
votes
1answer
117 views

Can you please check my proof of this limit of derivative

I proved the following: If $f$ is differentiable on an interval containing $0$ and if $\lim_{x > \to 0} f'(x) = L$ then $f'(0) = L$. Please can you tell me if my proof is correct: By ...
3
votes
1answer
247 views

Equivalences of continuity, sequential convergence iff limit (S.A. pp 106 t4.2.3, 110 t4.3.2)

1. This post became too long, ergo I moved this here. 2. I questioned anew here. How does $\color{red}{(I) \implies (III)}$? This contradicts $a \le b \not \implies \Leftarrow a < b$. 3. ...
2
votes
4answers
100 views

$\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers of $x$ and $y$

Give a convincing argument that $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers $x$ and $y$. Could someone please explain how to prove this? I attempted to say ...
1
vote
3answers
72 views

Cesaro identity for Fibonacci numbers

I am stuck with the identity $$ F_{2n} = \sum_{k=1}^n \binom{n}{k} F_k, $$ which happens to be formula 80. I am using induction, but so far without too much result. $$ \sum_{k=1}^{n+1} ...
8
votes
1answer
876 views

Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$

As a homework question, we got asked the following: Construct a function $f:[0,1] \rightarrow \mathbb{R}$ which is not the pointwise limit of any sequence of continuous functions Thinking about ...
7
votes
1answer
370 views

$\frac{dS}{d\rho}$ Factor arising

To get details see: equations 29,30,31,34,44,50,51 We have known some solitary wave solutions, given by(equations 1 to 5) $$ \phi_1=p_1\cos \tau \tag{1}$$ $$\phi_2=\frac16 ...
14
votes
3answers
363 views

If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$.

The proof is already given in the textbook but I tried other way around. Proof by contradiction: Let's assume that $p$ doesn't divide $a$ and $p$ doesn't divide $b$, but $p$ divides $ab$. So ...
5
votes
4answers
8k views

How many different words can be formed using all the letters of the word GOOGOLPLEX?

How many different words can be formed using all the letters of the word GOOGOLPLEX? I tried answering this problem and came up with the formula $n!/a!b!c!$ where ...
4
votes
4answers
2k views

Show $\max{\{a,b\}}=\frac1{2}(a+b+|a-b|)$

I am tasked with showing that If $a,b\in \mathbb{R}$, show that $\max{\{a,b\}}=\frac1{2}(a+b+|a-b|)$ I think I can say "without loss of generality, let $a<b$." Then $b-a>0$ But also, ...
2
votes
2answers
346 views

The union of a sequence of countable sets is countable.

While working on the theorem below, I constructed the following proof: Theorem. If $\left\langle E_{n}\right\rangle_{n\in\mathbb{N}}$ is a sequence of countable sets, then $$ ...
3
votes
2answers
128 views

Finding $P$ such that $P^TAP$ is a diagonal matrix

Let $$A = \left(\begin{array}{cc} 2&3 \\ 3&4 \end{array}\right) \in M_n(\mathbb{C})$$ Find $P$ such that $P^TAP = D$ where $D$ is a diagonal matrix. So here's the solution: $$A = ...