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

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0
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2answers
36 views

Is $f_n=x^n$ weakly convergent in $(\mathscr C[0,1],\lVert\cdot\rVert_\infty)$?

This is part of an old preliminary exam in Analysis I am working through. For earlier parts of the problem I have already shown that $f_n$ does not converge in $(\mathscr C[0,1],\lVert\cdot\rVert_\...
3
votes
6answers
115 views

Proof that $n^2 - 5$ is not divisible by 8

The question is from one of the past exams in a course I am doing. I have gotten halfway through it but cannot figure out how to finish it off. So the first part was to prove that $4 \mid n^2 - 5 $ ...
1
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2answers
35 views

$G$ nilpotent group and $N\trianglelefteq G$ then $[N,G]<N$, attempt of the proof

I need help in proving this fact: Let be $G$ nilpotent group and $N$ a normal and non trivial subgroup. Then $[N,G]$ is a proper subgroup of $N$. My attempt: I know the following fact: Let be $H$ ...
1
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2answers
46 views

Given $C \subset A \subset X$, why is that $C$ is closed in $X$ if $A$ is closed, $C$ is open in $X$ if $A$ is open?

I want to understand a result discussed here : Subspace of a normal space Let $(X, \mathfrak{T})$ be a topological space. Given $C \subset A \subset X$, let $C$ be a closed set in $A$, then claim ...
1
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1answer
41 views

Is this a valid proof for the following: Let $A$ and $B$ be sets. Show in general that $\overline{A \times B} \neq \overline{A}\times \overline{B}$

I am trying to practice my proofs involving cartesian products of sets and was kind of stuck on this practice question which the text did not provide solutions to. The question is as follows: Let $A$ ...
1
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0answers
72 views

Verify $y=x^aZ_p\left(bx^c\right)$ is a solution to $y''+\left(\frac{1-2a}{x}\right)y'+\left[(bcx^{c-1})^2+\frac{a^2-p^2c^2}{x^2}\right]y=0$ Method #2

This question is a sequel to this previous question. As before, some background information is needed first as follows from my textbook: The standard form of Bessel's differential equation is $$x^...
0
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3answers
32 views

If $(X, \mathcal{T})$ has a countable subbasis, then it has a countable basis

Given $(X, \mathcal{T})$ a topological space. Let $\mathcal{S}$ be a subbasis on $(X, \mathcal{T})$ Claim: If $\mathcal{S}$ is countable, then $\mathcal{T}$ has a countable basis $\mathcal{B}$ ...
1
vote
2answers
87 views

Show that $\mathbb{Z}[i]/n\mathbb{Z}[i] $ is a field if and only if $n$ is a prime number and $n\neq a^2+b^2, a,b\in\mathbb{Z}$.

I have to show the following statement: $\mathbb{Z}[i]/n\mathbb{Z}[i]$ is a field if and only if $n$ is a prime number and $n\neq a^2+b^2, a,b\in\mathbb{Z}$. Let $\mathbb{Z}[i]/n\mathbb{Z}[i]$ ...
1
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3answers
55 views

Proof verification: Construction of $\{x_n\}_n \subset \mathbb{Q}$ and $\{y_n\}_n \subset \mathbb{Q^c} $ that both converge to a real number $x$.

Can anyone check my proof. Im new to writing proof and i'm not really confident about it. Any feedback is appreciated thank you. Show that there exists two sequence strictly increasing such that $\...
2
votes
0answers
36 views

Find $F'(x)$ where $F(t) := \int_{g(x)}^0 f(x,t)dt$

Let $f: \mathbb{R^2}\to \mathbb{R}$ be $C^1$ and $g:\mathbb{R}\to \mathbb{R}$ also $C^1$. Define $$F(t) := \int_{g(x)}^0 f(x,t)dt$$ Find $F'(x)$. What I did was first define $\Phi : \mathbb{R}\...
0
votes
0answers
19 views

Prove that $d$ and $d'$ generate the same topology on $M$.

I'm teaching myself topology using a book I found. One of the exercises are to prove the following: Let $(M,d)$ be a metric space, let $c$ be a positive real number, and define a new metric $d'$ ...
0
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0answers
33 views

Proof verification: a set of $10$ times but not $11$ times differentiable functions is not a vector space

I need to find a counterexample showing that the set of $10$ times but not $11$ times partially differentiable functions is not a vector space (under the usual $+$ operator for functions and usual ...
1
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2answers
59 views

How to prove the power set of the rationals is uncountable?

Recently a professor of mine remarked that the rational numbers make an "incomplete" field, because not every subsequence of rational numbers tends to another rational number - the easiest example ...
3
votes
1answer
33 views

Construct a function g which increases faster than all $f_n$ for $n \in \Bbb{N}$

Let $f_n : \Bbb{N} \rightarrow \Bbb{N}$ $(n \in \Bbb{N})$ be a fixed collection of functions. Construct a function $g : \Bbb{N} \rightarrow \Bbb{N}$ such that for all $n \in \Bbb{N}$, $lim_{k \...
2
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0answers
53 views

Finding the structure of an $F_p[X]$module

$p$ is a prime and $M$ is an $F_p[X]-$ module. Given $(X-1)^3M=0, \vert (X-1)^2M\vert=p, \vert (X-1)M\vert=p^3$ and $\vert M\vert =p^7$, determine $M$ as an $F_p[X]-$module, up to isomorphism. Using ...
1
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1answer
62 views

Finite Union of Countable sets is countable

I have not studied the axiom of choice, I know how to prove that the union of two countable sets is countable and I want to use that a proceed by induction, but I'm not sure if my argument is okay. ...
2
votes
1answer
48 views

If $(f_n)\to f$ uniformly and $f_n$ is uniformly continuous for all $n$ then $f$ is uniformly continuous

Show if is true or false: if $(f_n)$ converges uniformly to $f$, and $f_n$ is uniformly continuous for all $n$ then $f$ is uniformly continuous I think is true. My attempt to prove it: if $(f_n)\to ...
1
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2answers
48 views

Proving the ratio of curvature and torsion is constant.

This question has been asked slightly differently in a few different forums, but I wanted to discuss my approach and see if I was on the right track: Prove that if the tangent lines of a curve make a ...
3
votes
0answers
42 views

My proof that $f^{-1}(D_1 \cap D_2) = f^{-1}(D_1) \cap f^{-1}(D_2)$

I'm currently self studying proof and set-theory, and I'm quite new to both of them. As an exercise, I'm practicing proving some basic theorems, so it'll be great if you can give me some feedback on ...
1
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2answers
48 views

Proof concerning regular space: there exist a closed set contained in any open set containing $x$

I was given a claim: Let $(X, \mathfrak{T})$ be a topological space. Then $X$ is a regular space iff $\forall x \in X, \forall U \in \mathfrak{T}$ s.t. $x \in U$, $\exists V$ such that $x \in ...
2
votes
1answer
20 views

Prob. 4 (b), Sec. 20 in Munkres' TOPOLOGY, 2nd ed: Which of these sequences are convergent w.r.t. the product, uniform, and box topologies?

Let $\mathbb{R}^\omega$ denote the set of all the (infinite) sequences of real numbers. Then which of the following sequences in $\mathbb{R}^\omega$ are convergent (and if so, then to which points(s)) ...
0
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0answers
39 views

Prove Exponential series from Binomial Expansion

I try to prove the Exponential series : $$\exp(x) = \sum_{k=0}^{\infty} \dfrac{x^k}{k!}$$ From the definition of the exponential function $$\exp(x) \stackrel{\mathrm{def}}{=} \lim_{n\to\infty} \left(...
0
votes
2answers
50 views

Integral of $\int{x+3\over (3-2x)^{2\over 3}}$

I am finding the ingetral of: $$\int{x+3\over (3-2x)^{2\over 3}}$$ So, $u=3-2x$, $_{}$ $du=-2 dx$,$_{}$ and $x={1\over 2}(u-3)$ $$\int{{1\over 2}(u-3)+3 \over u^{3\over 2}}({-{1\over 2}})du \\ = {...
3
votes
0answers
31 views

Strict inequality in Fatou's lemma

Put $f_n={1}_E$ if $n$ is odd, $f_n=1-1_E$ if $n$ is even. What is the relevance of this example to Fatou's lemma? Proof: We see that $\int \limits_{X}f_nd\mu= \mu(E^c)$ if $n$ is even and $\int \...
26
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2answers
560 views

A novelty integral for $\pi$

My lab friends always play a mentally challenging brain game every month to keep our mind running on all four cylinders and the last month challenge was to find a novelty expression for $\pi$. In ...
0
votes
1answer
45 views

a countable and connected metric space $X$ has only one point

Prove that a countable and connected metric space $X$ has only one point. Could you check my attempt? Thanks! Suppose not. Consider two distinct point $x$ and $y$. Define $d:=d(x,y)>0$. Let $a<...
0
votes
1answer
54 views

Prove that: $ \int_{a}^{a+T}f(x)dx=\int_{0}^{T}f(x)dx$

Let $f:R\rightarrow R$ with a period of $T\gt0$, and $f(x)=f(x+T)$ for every x. Assume that $f$ is integrable on $[0,T].$ 1.) Prove that $f$ is integrable on $[a,a+T]$ 2.) Prove that: $$ \int_{a}^{a+...
4
votes
2answers
85 views

Show that $\lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n} x_k=x $ if $\{x_n\} \to x$

If $\ \{x_n\}$ converge to $x$ show that: $$\lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n} x_k =x.$$ Let choose an $\epsilon>0$, then $\exists N_1\in \mathbb{N}$ such that $\forall n\geq N_1$ we ...
3
votes
4answers
137 views

Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$

Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{n}{\left(n+1\right)!} = 1-\frac{1}{\left(n+1\right)!}$ So I proved the base case where $n=1$ and got $\frac{1}{2}...
1
vote
0answers
34 views

If for each $\alpha, \beta$, the map $\psi_{\beta} \circ F \circ \phi_{\alpha}^{-1}$ is smooth, then $F$ is smooth.

Let $F:M\to N$ be a map. Suppose $\mathcal{A} = \{(U_{\alpha},\phi_{\alpha})\}$ and $\mathcal{B} = \{(V_{\beta},\psi_{\beta})\}$ are smooth atlas for $M$ and $N$ respectively. Suppose that for each $\...
2
votes
2answers
71 views

Integral of $\int{(x^2+2x)\over \sqrt{x^3+3x^2+1}} dx$

Find the integral of the following: $$\int{(x^2+2x)\over \sqrt{x^3+3x^2+1}} dx$$ Do set $u=x^3+3x^2+1$? So, $du=(3x^2+6x)dx$? And, $x^2+2x={u-1-x^2\over x}$? So then, $$\int{({u-1-x^2\over x})\...
1
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1answer
47 views

Any subring of $A$ is an ideal. If $A$ is an integral domain then $A$ is commutative

Any subring of $A$ is an ideal. If $A$ is an integral domain then $A$ is commutative. Is my proof correct? So let $a$ and $b$ nonzero elements of $A$. $C(a)=\{ x\in A \mid ax=xa\}$ Is a subring ...
0
votes
1answer
28 views

How to show a continuous function from a space to a subspace is continuous from a space to the whole space?

Let $(X,\mathcal{T})$ and $(Y, \mathcal{J})$ be topological spaces. Let $W \subset Y$ be a subspace with its subspace topology. Show that if $f: X \to W$ is a continuous function, then $f: X \to ...
0
votes
2answers
41 views

Can someone tell me how to prove $\cap_{i\in I}(B \setminus A_i) = B \setminus (\cup_{i\in I} A_i)$ [closed]

Can someone tell me how to prove $\cap_{i\in I}(B \setminus A_i) = B \setminus (\cup_{i\in I} A_i)$
1
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1answer
26 views

Closure of a set is closed - proof verification

I want to prove the following: Let $(X, \mathcal{T})$ be a topological space. Let $A \subset X$ be a subset. Then $\mathrm{cl}A$ is closed. Our definition is $\mathrm{cl}A = \{x \in X \mid \...
0
votes
1answer
42 views

Confusion on the definition of ∩i∈I Ai

Having a hard time with this one. As I understand it ∩F equals $$\{ x: ∀A ∈ F, x ∈ A)\}$$ which is also equivalent to $\underset{i\in I}{∩} A_i$. And this would mean if F were {{1,2,4},{4,7,8}} ...
1
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1answer
31 views

Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$ under some conditions

Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$, with $f: (k_1,k_2)\to\Bbb R$ differentiable at $a\in (k_1,k_2)$, $\lim b_n=\lim a_n=a$, and $a_n<a<b_n$. My attempt: Im ...
1
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1answer
37 views

Convolution of PDFs is a PDF

Suppose $f$ and $g$ are PDFs of real-valued random variables. Show that the convolution $f\ast g$ of $f$ and $g$ (defined below) is also a PDF. $$(f\ast g)(x)=\int_{-\infty}^\infty f(y)g(x-y)\,dy.$$ ...
2
votes
2answers
33 views

Write the contrapositive of an if-then statement

$\forall a, a' \in A,$ if $f(a)=f(a'),$ then $a=a'$ Here is my attempt: $\exists a, a' \in A,$ if $\sim (a=a')$, then $\sim (f(a)=f(a'))$ Did I attempt to do this correctly? I based this on the ...
1
vote
1answer
45 views

Classifying groups of order 6

I'm trying to proof that if a group $G$ has order $6$, then it is either $\mathbb{Z}_{6}$ or $S_{3}$. I know that there are a lot of solutions to this on the internet, but I want to know why I found ...
0
votes
2answers
43 views

For all real numbers x and y there is a real number $z$ such that $x + z = y − z$.

To Prove: For all real numbers $x$ and $y$ there is a real number $z$ such that $x + z = y − z$. Proof: $x+z=y-z \Rightarrow y-x=2z$. Since $y$ and $x$ are real numbers, $2z$ is also real. Therefore ...
0
votes
1answer
44 views

For which values of $\gamma$ does this inequality hold?

Edited: Just realised my first post was somewhat misleading and not precise. Thanks to the two commetators that pointed it out. I am working on an article and ended up wondering for which values of $\...
1
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0answers
19 views

Proposition 2.3 of Quiver Representations by Ralf Schiffler

I'm trying to prove proposition 2.3 of Quiver Representations by Ralf Schiffler. To any vertex $i$ finite acyclic quiver $Q$ we can associate the indecomposable projective $P(i)$, this proposition ...
0
votes
2answers
25 views

Show that the closure of $A$ is the intersection of all closed sets containing $A$, topology proof needed

I want to show that given $(X, \mathcal{T})$, we define $\overline A = \{x \in X| \forall U \in \mathcal{T}, x \in U \implies U \cap A \neq \varnothing\}$ (definition of closure from Munkres), then ...
1
vote
2answers
58 views

Find the sum of all 4-digit numbers formed by using digits $0, 2, 3, 5$ - possible formula for competitive exam

Find the sum of all 4-digit numbers formed by using digits 0, 2, 3, 5 without repetition There is a similar question in this site and Eric Tressler has provided a clear method to solve such ...
1
vote
1answer
73 views

Prove the formula is a contadiction

For every $A$ in Propositional calculus and for every $\rho$ we define: $$A^\rho = \begin{cases} A & \text{if $[|A|]_\rho = true$} \\ \lnot A & \text{if $[|A|]_\rho = false$} ...
0
votes
0answers
10 views

PDE $Ru_{zzz}+u_{zzw}=0$: Solution verifcation

I tried to solve this PDE $$Ru_{zzz}+u_{zzw}=0$$ First i substituted $u_{zz}=v$: $$Rv_{z}+v_{w}=0$$ I solved this by the method of characteristics and integrated the result with respect to $z$ ...
1
vote
1answer
36 views

Proving the column-row expansion of two matrices $A$ and $B$ is equal to the product $AB$

I'm fairly new to proofs with matrices, so determining when I'm allowed to use certain indices to denote arbitrary entries in various matrices is still a challenge. For this problem, I want to prove ...
4
votes
3answers
100 views

Property of odd ordered elements of a Group

I'm (slowly) working my way through "Abstract Algebra" by Dummit and Foote. In the first set of exercises on group theory, the following question is posed: "Let $G$ be a finite group and let $x$ be ...
1
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1answer
38 views

Real Analysis Folland Proposition 1.13

Proposition 1.13 - If $\mu_0$ is a premeasure on $\mathcal{A}$ and $\mu^*$ is defined by (1.12) then a.) $\mu^*|\mathcal{A} = \mu_0$ b.) every set in $\mathcal{A}$ is $\mu^*$-measurable ...