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

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14
votes
4answers
1k 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 ...
1
vote
1answer
148 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 ...
1
vote
0answers
48 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 ...
2
votes
2answers
6k 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
245 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 ...
5
votes
1answer
181 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
1answer
284 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 > ...
2
votes
1answer
888 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 ...
13
votes
4answers
2k 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 ...
2
votes
1answer
172 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 ...
1
vote
2answers
295 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
189 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
2answers
180 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 ...
6
votes
5answers
590 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) ...
3
votes
2answers
62 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 ...
3
votes
1answer
286 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
1answer
49 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 ...
2
votes
1answer
71 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 ...
2
votes
5answers
206 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] & = ...
2
votes
2answers
177 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 ...
0
votes
1answer
69 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
65 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 ...
6
votes
2answers
680 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
244 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 ...
3
votes
2answers
2k 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: ...
1
vote
0answers
226 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 ...
5
votes
4answers
143 views

Evaluating $\lim_{n \to \infty} (1 + 1/n)^{n}$ [duplicate]

I was recently thinking about how I could evaluate the famous limit of 'e' as I haven't ever seen a proof. I can't really find anything online so I've tried to evaluate the limit myself. And I was ...
4
votes
1answer
177 views

let $H\subset G$ with $|G:H|=n$ then $\exists~K\leq H$ with $K\unlhd G$ such that $|G:K|\leq n!$ (Dummit Fooote 4.2.8)

Question is to prove that For $H\subset G$ with $|G:H|=n$, $\exists~K\leq H$ with $K\unlhd G$ such that $|G:K|\leq n!$ What i have done so far is that : $H$ be a subgroup of index $n$ in $G$ and ...
3
votes
2answers
48 views

A subsequence of a convergent sequence converges to the same limit. Questions on proof. (Abbott p 57 2.5.1)

Solutions to Homework 3 doesn`t duplicate. We have to prove that if $(a_{n})$ is a sequence in $\mathbb{R}$ with $\displaystyle \lim_{n\rightarrow\infty} a_n =a$, and if $(a_{n_{k}})_{k\in ...
2
votes
1answer
113 views

Question about integration on a box

Let $Q \subseteq \mathbb{R}^n$, and $f: Q \to \mathbb{R} $ is integrable over $Q$. $f \geq 0$. if $A \subseteq Q$, then $\int_Q f \geq \int_A f $ Attempt: say $\epsilon > 0$ Let $P_1$ be a ...
1
vote
4answers
400 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
1answer
90 views

Permutation proofs

I have just started going though "An Introduction to the theory of groups" by J.J Rotman. I have questions above the following two exercises: " The identity function $1_X$ on a the set $X$ is a ...
12
votes
3answers
1k views

Show that the product of two consecutive natural numbers is never a square.

I'd like to have my proof verified and if possible, to see other solutions that are interesting. Proof: Suppose $n(n+1)$ is a square. Then we write $$n(n+1) = \prod_{p} p^{c(p)}$$ where $c(p) = a(p) ...
5
votes
2answers
558 views

Proof to sequences in real analysis

I need some verification for my proof in part a) and help to get me started on part b) a) Prove that the sequence $a_n = (2n+1)/(3n+5)$ converges to $2/3$ directly from the definition of convergence ...
5
votes
3answers
1k views

Strong Induction Proof: Fibonacci number even if and only if 3 divides index

The Fibonacci sequence is defined recursively by $F_1 = 1, F_2 = 1, \; \& \; F_n = F_{n−1} + F_{n−2} \; \text{ for } n ≥ 3.$ Prove that $2 \mid F_n \iff 3 \mid n.$ Proof by Strong Induction : ...
4
votes
1answer
219 views

The product of any three consecutive natural numbers is divisible by 9.prove or find a counterexample

can anyone give me feedback on my solution false counterexample: n(n+1)(n+2) let n= 1 1(2)(3) =6 is not divisible by 9 is this correct?
4
votes
2answers
76 views

Integration of Rational Functions - Problem with proof relating to complex solutions

$\quad$I was reviewing integration of rational functions, and all was going well until I saw this bit, in the end of the explanation: (translated from portuguese by me) $ \qquad \qquad\text{with ...
3
votes
1answer
75 views

Without Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = (-y^3,x^3,z^3)$ - 2012 9C

Question: 2012 9C. Consider the (cutoff) paraboloid defined by $z= x^2 + y^2 , \frac{1}{9} \le z \le 1$. Sketch the surface. Verify Stokes’s Theorem for for $\mathbf{F} = (-y^3,x^3,z^3)$. Herein, I ...
3
votes
2answers
107 views

If $f$ is differentiable and $\lim_{x→0} f'(x) = L$, then $f'(0) = L$.

True/False. (c) If $f$ is differentiable on an interval containing zero and if $\lim_{x→0} f'(x) = L$, then $f'(0) = L$. 1. How to presage proof by contradiction? Proof by contradiction. ...
2
votes
3answers
68 views

Monotone Union of subgroups being subgroup

I saw this exercise in a book: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, if $\{H_t\}$ is a monotonic collection, show that ...
2
votes
2answers
292 views

When Dim eigenspace = 1, any $2\times 2$ complex matrix A is similar to $\left(\begin{array}{ll} \lambda & 1\\ 0 & \lambda \end{array}\right)$.

$\bbox[5px,border:2px solid gray]{ \text{ Case 3 } }$ If $\dim E_{\lambda}=1$, take a nonzero $v\in E_{\lambda}$, then $\{v\}$ is a basis for $E_{\lambda}$. Extend this to a basis $\mathfrak{B}=\{v,\ ...
2
votes
2answers
81 views

Modus operandi for proving Evaluation Fundamental Theorem of Calculus (Abbott p 200, Spivak p 272 T14.2)

1. How can we presage to use Mean Value Theorem to start the proof? 2. Mean Value Theorem engenders a point in an open interval. Shouldn't this be $x_i \in (t_{i - 1}, t_i) $? After ...
2
votes
0answers
83 views

Choices of epsilons in proof : $(b_n) \to b$ implies $\left\{\frac{1}{b_n}\right\} \to \frac{1}{b}$ (Abbott pp 47 T2.3.3.iv) [closed]

Original became long, ergo I ask anew. The trick is to look far enough out into the sequence $(b_n)$ so that the terms are closer to b than they are to 0. Consider the particular value $e_0 = |b|/2$. ...
2
votes
1answer
23 views

Reduce all cases to $x \to 0^{+}$ and $f(x),g(x) \to 0$ before proving L'Hôpital's Rule

Hypotheses: I. $f$ and g differentiable on $(a,b)$ and continuous on $(a,b]$, II. $\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0 \quad $ or $\pm \infty$ III. $\lim_{x \to a} f'(x)/g'(x)$ exists IV. ...
2
votes
0answers
141 views

Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles.

I would like to know if my proof below is correct. Problem Let $p$ be a prime. Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles. Show by ...
2
votes
3answers
773 views

The power set of the intersection of two sets equals the intersection of the power sets of each set

It would be great if someone could verify this proof. Theorem: $\mathcal{P}(A \cap B) = \mathcal{P}(A) \cap \mathcal{P}(B)$ Proof: First I prove that $\mathcal{P}(A \cap B) \subseteq ...
2
votes
0answers
140 views

Hatcher's Algebraic Topology 0.6(a) - Is this proof legit?

I am primarily not sure about my two-step homotopy construction: Let $X$ be the subspace of $\mathbb{R}^2$ consisting of the horizontal segment $[0, 1] \times \{0\}$ together with all the ...
1
vote
0answers
38 views

proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian

If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian. Is that statement correct? Here's my attempt of proof: Let $a,b \in G$, then: ...
1
vote
1answer
73 views

Chain rule (proof verification)

Hi everyone I'm asking two thinks is this proof correct (my other idea was using limit of sequences)? and are there a simpler alternative than this using Newton's approximation? If someone could help ...
1
vote
2answers
94 views

Using induction, prove that $(-7)^n -9^n$ is divisible by $16$

First of all, I think the problem should be $(-7)^n -9^n$ is divisible by $-16$ because if I test the basis by letting $n=1$, I have $-16$ instead of $16$. Edit: Alright ... I sort of understand why ...