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

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20
votes
4answers
2k 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 ...
4
votes
2answers
122 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 ...
13
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 ...
6
votes
2answers
1k 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
1answer
815 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 > ...
1
vote
1answer
229 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
5answers
324 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] & = ...
22
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 ...
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 ...
3
votes
3answers
523 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 ...
1
vote
3answers
811 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 ...
4
votes
4answers
801 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 ...
1
vote
0answers
58 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 ...
8
votes
5answers
1k 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) ...
2
votes
2answers
14k 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 ...
7
votes
2answers
432 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 ...
3
votes
1answer
226 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. ...
3
votes
1answer
3k 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
1answer
90 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 ...
6
votes
2answers
132 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 ...
2
votes
1answer
358 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 $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ ...
4
votes
0answers
151 views

6-digit password - a special decoding method

Consider the situation of decoding a 6-digit password that consists of the symbols A to Z and 0 to 9, where all possible combinations are tried randomly and uniformly. Consider the ...
3
votes
1answer
589 views

Ideals in ring of continuous functions $\mathcal{C}[0,1]$ … NBHM- Algebra

I would like to compile all questions I have encountered with Ideals in the ring $\mathcal{C}[0,1]$ of all continuous real valued functions and ask if there are any gaps. Question is to see if : ...
2
votes
3answers
122 views

Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A =\displaystyle{ \sum_{i=1}^n} f_0(x_1, \ldots, Ax_i,\ldots, x_m)$

Question: Consider $f: (-\epsilon, \epsilon) \to \mathbb R^{m^2}$ a differentiable path of matrices $m \times m$ such that $f(0) = I_m$ and the function $g: I \to \mathbb R$ is defined by $$g(t) = ...
2
votes
1answer
191 views

Sufficient and necessary conditions to get an infinite fiber $g^{-1}(w)$

I want to verify the proof of this result and get some start ideas to overcome the different steps of this proof. Lemma: Let $g$ be a real analytic function. Then we have the equivalence ...
2
votes
2answers
238 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 $$ ...
1
vote
4answers
75 views

Solution verification: Prove by induction that $a_1 = \sqrt{2} , a_{n+1} = \sqrt{2 + a_n} $ is increasing and bounded by $2$

I have the following recursive relation (sequence): \begin{align} a_1 = \sqrt{2}, \quad a_{n+1} = \sqrt{2 + a_n} \end{align} My Try: I'm a little skeptical of my manipulations near the end but it ...
1
vote
2answers
823 views

Using inequalities and limits

Is it possible to say: $$ If \ f(x) \ and \ g(x) \ both \ have \ limits \ as \ x\to p\ and \ f(x) \le g(x), \ then \lim_{x \to p} f(x)\le \lim_{x \to p} g(x). $$ My proof(Edit: Proof is wrong due to ...
1
vote
3answers
391 views

homomorphism from $S_3$ to $\mathbb Z/3\mathbb Z$

TRUE/FALSE TEST: There is a non-trivial group homomorphism from $S_3$ to $\mathbb Z/3\mathbb Z.$ My Attempt: True: Choose $a,b\in S_3$ such that $|a|=3,|b|=2.$ Then ...
6
votes
3answers
1k 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 ...
4
votes
1answer
473 views

Proof of equicontinuous and pointwise bounded implies compact

I tried to prove the Arzela-Ascoli theorem: Let $X$ be a compact Hausdorff space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb R$ endowed with the sup norm $\|\cdot ...
4
votes
2answers
355 views

Questions on Proofs - Equivalent Conditions of Normal Subgroup - Fraleigh p. 141 Theorem 14.13

(1.) Why did Fraleigh shirk the proof for $(2) \implies (1)$? By dint of Arthur's comment, $(2) \iff \color{crimson}{gHg^{-1} \subseteq H} \quad \wedge \quad gHg^{-1} \supseteq H \implies ...
3
votes
0answers
408 views

Vector spaces - Multiplying by zero scalar yields zero vector

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space related axioms. ...
3
votes
3answers
368 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 ...
2
votes
2answers
550 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 ...
2
votes
2answers
211 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 ...
1
vote
3answers
78 views

Proof that a sequence is convergent

I'm asked to prove the convergence of the sequence $$X_n=\left(1+\frac12\right)\left(1+\frac14\right)\left(1+\frac18\right)\cdots\left(1+\frac{1}{2^n}\right)$$ I proved that it is increasing through ...
0
votes
4answers
77 views

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

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
888 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 ...
0
votes
1answer
83 views

Any $2\times 2$ complex matrix A is similar to one of these three: (See first line of the question)

(i) : $\left(\begin{array}{ll} \lambda_{1} & 0\\ 0 & \lambda_{2} \end{array}\right)$, (ii) : $\left(\begin{array}{ll} \lambda & 0\\ 0 & \lambda \end{array}\right)$, (iii) : ...
0
votes
1answer
76 views

Linear Congruence Theorem - Are these solutions too? Where'd they hail from?

(1) Can't the signs - I colored them in red - of x and y be switched? Aren't $x = x_0 - bn/d$ and $y = y_0 + an/d$ also solutions? They satisfy $ax + by = c$? (2) How can I remember these ...
0
votes
1answer
97 views

Proof: $\sum_{i=1}^pE_i \doteq \bigoplus_{i=1}^p E_i \leftrightarrow \forall i\in \{1,…,p\}(E_i \cap \sum_{t \in \{1,…,p\}-\{i\}}E_t=\{0\})$

I am using the following definition: Def.: let $E_1,...,E_p$ $p$-vector subspaces of $V$, $E_1+E_2+...+E_p$ is direct sum, $E_1+E_2+...+E_p \doteq E_1\oplus E_2 \oplus ... \oplus E_p$, if $$\forall ...
44
votes
12answers
6k 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 ...
11
votes
4answers
6k views

Prove that $\sqrt 5$ is irrational

I have to prove that $\sqrt 5$ is irrational. Proceeding as in the proof of $\sqrt 2$, let us assume that $\sqrt 5$ is rational. This means for some distinct integers $p$ and $q$ having no common ...
3
votes
1answer
426 views

Tricks. If $\{x_n\}$ converges, then Cesaro Mean converges (S.A. pp 50 2.3.11)

Show if $\{x_n\}$ is a convergent sequence, then the sequences given by the averages $\{\dfrac{x_1 + x_2 + ... + x_n}{n}\}$ converges to the same limit. (Not a duplicate) Let $\epsilon>0$ be ...
10
votes
2answers
155 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 ...
4
votes
4answers
647 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, ...
4
votes
2answers
3k views

Proving Cauchy's Generalized Mean Value Theorem

This is an exercise from Stephen Abbott's Understanding Analysis. The hint it gives on how to solve it is not very clear, in my opinion, so I would like for a fresh set of eyes to go over it with me: ...
3
votes
0answers
363 views

Hatcher's Algebraic Topology Problem 0.6bc - is this proof legit?

I am tackling Hatcher's Algebraic Topology Problem 6b. But I am wonder if my proof is okay, or that is too descriptive and I need to carry out explicit computation in coordinates? Also, I am not sure ...
1
vote
4answers
63 views

Wilson's Theorem textbook proof question

I'm trying to understand this proof from Stein's Elementary Number Theory, and I understand the pairing of inverses but not the other direction. I have two questions: $1).$ When the proof says, $l$ ...