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

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17
votes
5answers
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 ...
2
votes
5answers
271 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] & = ...
3
votes
1answer
543 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
187 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 ...
3
votes
2answers
72 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 ...
1
vote
0answers
54 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 ...
6
votes
2answers
908 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 ...
2
votes
2answers
10k 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
269 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 ...
7
votes
5answers
819 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) ...
5
votes
1answer
198 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
2k 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 ...
14
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 ...
4
votes
0answers
124 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 ...
2
votes
1answer
184 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
595 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
281 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
300 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
174 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
1answer
445 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
60 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
72 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
2answers
190 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
74 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
1answer
79 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
82 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 ...
5
votes
1answer
336 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 ...
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: ...
2
votes
0answers
289 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 ...
13
votes
2answers
495 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 ...
11
votes
4answers
729 views

Scratch work for delta-epsilon proof for $\lim_{x \to 13} \sqrt{x-4} = 3$

Prove $\lim_{x \to 13} \sqrt{x-4} = 3$. We need to show for all $E> 0$ there exists $D > 0$ such that if $0 < |x - 13| < D$, then $|\sqrt{x-4} - 3| < E$. Let me write D for ...
5
votes
4answers
146 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
208 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
1answer
72 views

Proof of the First Isomorphism Theorem for Groups

The statement is the First Isomorphism Theorem for Groups from Abstract Algebra by Dummit and Foote. This proof was left as a exercise, so I'd like to check if all is ok. In particular, I'm a bit ...
3
votes
2answers
104 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
3answers
133 views

Prove that $ 2^n \not \equiv 1 \pmod{n} $ for any $n > 1$.

I have proved this in following way. Assume that $ 2^n \equiv 1 \pmod{n} $. that means $n\mid(2^n -1)$. but by proof by contradiction, for $n=3$ this does not hold and we can say $n \nmid (2^n-1) ...
2
votes
1answer
115 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 ...
2
votes
3answers
105 views

Prove $\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$

I need to prove the following: If $n,m,k\in \mathbb{N}$ and $k\leq m \leq n$, then $$\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$$. I did the following steps: \begin{align} ...
1
vote
3answers
212 views

Proving commutativity of addition for vector spaces

I'm trying to prove commutativity of addition for vector spaces, using the axioms for vector spaces. Apparently commutativity can be proven! Im having trouble getting a good feel for what is allowed ...
1
vote
4answers
663 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
97 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 ...
0
votes
1answer
61 views

Integrability condition on the Fourier transform implies that the function is infinitely differentiable

This is a problem from the book from Stephane Mallat "A wavelet Tour of signal processing: a sparse way". A function $f$ is bounded and $p$ times continuously differentiable with bounded derivatives ...
12
votes
3answers
2k 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
752 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
2k 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
2answers
90 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
5answers
511 views

Proof of uniqueness of identity element of addition of vector space

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. Proof of uniqueness of identity element of addition of vector space ...
3
votes
3answers
486 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 ...
3
votes
1answer
64 views

Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$ [duplicate]

Can someone please verify this? Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$ let $x \in \mathcal{P}(B)$. Then, $x \subseteq B$ This implies that $$\forall a \in ...
3
votes
1answer
77 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 ...