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

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4
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1answer
40 views

Show that if $(\sum x_n)$ converges absolutely and $(y_n)$ is bounded then $(\sum x_n y_n)$ converges

This is the exercise 2.7.6 of the book Understanding analysis of Abbott, I want a check of my proof and if is needed additional information to complete it. a) Show that if the sequence $(\sum ...
1
vote
0answers
20 views

Regular values of $g(x,y)= x^2 - y^2$

I am doing some very introductory studying about manifolds. I wanted to check I was getting the right end of the stick through this example. Could anyone verify/correct my solution to the following ...
0
votes
1answer
13 views

Is the weak-* topology on a topological vector space Hausdorff?

Let $V$ be a topological vector space and $V^*$ be the space of linear functionals induced with the weak-* topology. Can we say that $V^*$ is Hausdorff? Here is my attempt: Let $\lambda\ne\lambda'\in ...
1
vote
0answers
33 views

Details on Proving that $\lim_{n \rightarrow \infty}\int_{-M}^M f(x) \cos (nx) dx=0$ Using Density of Step Functions

I was working on a question very similar to this post: Show that $\int_{-\pi}^\pi ~f(x) \cos (nx) \mathrm{d}\mu(x)$ converges to $0$ . I want to show that $\lim_{n \rightarrow \infty}\int_{-M}^M ...
1
vote
1answer
25 views

If $X$ is totally bounded then every sequence contains a Cauchy subsequence

I attempted the proof, I just want to see if it is correct: Suppose $X$ is totally bounded and $(x_n)$ is a sequence in $X$. Then $(x_n)$ has a subsequence contained in a ball of radius $1/2$. This ...
0
votes
2answers
25 views

Check my proof on showing a graph with each vertex's degree at least $e$ has every tree with $e$ edges a subgraph

Let $T$ be a tree with $e$ edges and $G$ be a simple graph such that ech vertex has degree at least $e$. We need to show that $T$ is a subgraph of $G$. I tried to prove this by induction. The base ...
0
votes
1answer
32 views

Is the conjunction of all necessary statements sufficient? What about the converse?

A necessary condition for consequent $q$ is a proposition $p$ such that: $$\neg p \implies \neg q$$ let $P:= \{p_i: \neg p_i\implies \neg q\}$ What I want to know is if $$\bigwedge_{p_i\in P} p_i ...
1
vote
1answer
41 views

Prove $n < 2^n$ for all $n \geq 0$ using induction.

Please verify for me? Base case: $n = 0 $ $0 < 2^0$ $0 < 1$. This is true. Inductive step: Suppose $n \geq 0$. Assume $P(k)$ is true if $k = n$. We must deduce that $P$ holds for $k+1$. $n ...
0
votes
0answers
44 views

Topology bases for $\mathbb{R}_{\text{usual}}$

I'm trying to compile correctly formulated solutions to common topology questions as a summer project. I'm not very confident in my proof writing abilities so I'm going to post my solutions here for ...
0
votes
1answer
32 views

In the co-finite topology and the co-countable topology, must $X$ be finite or countable?

Recall $\tau_{co-finite} = \{U \subseteq X| X\backslash U \text{ is finite}\}\cup\{\varnothing\}$ $\tau_{co-countable} = \{U \subseteq X| X\backslash U \text{ is countable}\}\cup\{\varnothing\}$ ...
1
vote
2answers
44 views

Does the Hausdorff property hold on closed subsets of $\mathbb{R}^n?$

I am trying to prove that given disjoint closed $A,B\subseteq \mathbb{R}^n$, there exist disjoint open $U,V$ containing $A,B$ respectively. In other words that we can take the Hausdorff property to ...
2
votes
5answers
58 views

Proving that the groups $S_3$ and $D_6$ are isomorphic [duplicate]

In particular, $S_3$ is the group of permutations of $\{1,2,3\}$, and $D_6$ is the dihedral group of symmetries of the triangle (written as $D_{2\cdot 3}$). In generator-relation form, $D_6 = ...
1
vote
0answers
12 views

Spectral Representation of the Green's Function

I have been given an exam question asking to show that the spectral representation of the Green's Function for a given Hermitian Operator admits solutions of a given form. I have an answer, but the ...
0
votes
1answer
15 views

Separable infinite-dimension Hilbert space and its subspaces

Suppose $H$ is any separable infinite-dimensional Hilbert space. Then $H$ has family of closed subspaces $\big\{ E_t :~ t \in [0,1]\big\}$ such that $E_s$ is a strict subspace of $E_t$ for all $0 ...
1
vote
2answers
40 views

Constructing topology on $\Bbb{Z}$

Fix an infinite subset $A$ of $\mathbb Z$ whose complement $\mathbb{Z}\setminus A$ is also infinite. Construct a topology on $\mathbb{Z}$ in which: (a) $A$ is open (b) Singletons are never open (i.e ...
0
votes
0answers
21 views

$\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$

$$\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$$ traversing the unit circle counterclockwise. So the singularities are $z=0$ and $z=-2$ but the second is outside the unit circle so it isn't ...
0
votes
1answer
45 views

Gauss-Legendre three point rule

Use the change of variables $$x=\frac{a+b}{2}+\frac{b-a}{2}t,$$ to show that $$\int^b_a f(x) \ dx = \frac{b-a}{2} \int^1_{-1} f\left( \frac{a+b}{2} + \frac{b-a}{2}t \right) \ dt.\tag{1}$$Hence ...
1
vote
0answers
25 views

$p \in \Bbb P, a \in \Bbb N$, then if $ord_p(a)=d$ we have $a^{d-1}+\dots+a+1 \equiv 0 \mod p$.

I want to prove the statement in the title, but I think we need $d \geq 2$ in the statement since otherwise there is a case not fulfilling the statement. My attempt: By assumption we have ...
0
votes
1answer
14 views

If $d(x_n, x_{n+k}) < \epsilon$ for all $n \ge N\in \mathbb{N}$, and $k \in \mathbb{N^*}$, then $(x_n)$ is a Cauchy sequence?

I've tried to prove this is as: If the condition holds for any k, then $\sum_{i=1}^{\infty} d(x_i, x_{i+1}) < \infty$, by the comparison test ($\epsilon < 1$). Then the sequence of the partial ...
1
vote
2answers
35 views

How is f(4,4,4)=48 a local minimum? Can it be inferred that it is either a maxima or minima & only one extreme value within the constraint?

Disclaimer: In the definition (Stewart Calculus, 7E): "Method of Lagrange Multipliers" part (b)- Evaluate $f$ at all extreme points $(x,y,z)$ from step a. The largest of these values is the maximum ...
3
votes
0answers
34 views

Isomorphism of Non-Symmetric Matrix when Permutation-Set is given: A simple observation

Context: Consider, two $m \times n$ matrices $A, B$ such that there is a permutation $\kappa$ that such that such that $A^{\kappa}=B$ (Wielandt's notation), i.e. $A, B$ are isomorphic but not ...
2
votes
2answers
63 views

Can we define a metric on $Y$ such that all continuous mappings $f:X\rightarrow Y$ are constant?

Given that $Y$ contains more than one element and let $X$ be the real line equipped with the standard metric. Then can we define a metric $\sigma$ on $Y$ such that every continuous mapping ...
0
votes
1answer
23 views

$\mathbb{Z}\setminus U$ is open, where U is a basic open set of $\mathcal{B}$, the set of all arithmetic progressions

Let $m, b \in \mathbb Z$ with $m \neq 0$, and $U$ is of the form $Z(m, b) = \{ mx + b \mid x \in \mathbb Z \}$ I'm not sure how to show $\mathbb{Z}\setminus U$ is open, I was thinking to expressing ...
0
votes
0answers
9 views

What is the fastest way to show that FT Convolution theorem holds also in the case of a weighted sum?

Given $Y=X+Z$, with $X, Z $ r.v. such that $X \sim f(x)$ and $Z \sim g(x)$, the Fourier Transform Convolution property gets me the result: $$\mathcal{F}[(f \otimes g)(x)] = \hat{f}(\xi) \hat{g}(\xi) ...
1
vote
1answer
28 views

Meeting point in a circular race

X and Y walk around a circular course 100 km from the same point. If they walk at 5 kmph and 7 kmph respectively in the same direction when will they meet? Answer according to ...
2
votes
2answers
266 views

Gnarly equality proof? Or not?

I have this gnarly equality which Mathematica's Reduce says it doesn't have the chops to handle: $\left.k\in \mathbb{Z}\land \left\lceil \frac{k-2}{2}\right\rceil ...
0
votes
1answer
61 views

Find all real solutions of $ \frac{ae^x}{2e^x-1} < 1 $

Question: Find all real solutions of $ \frac{ae^x}{2e^x-1} < 1 $, where $a$ is a positive constant. This is what I have attempted: Consider $$ \frac{ae^x}{2e^x-1} < 1 $$ Case 1: ...
3
votes
2answers
37 views

Could anybody please check my proof about connected graph?

I have written a proof of the following statement but not sure whether it is correct or not. Let $G$ be a connected graph and each vertex has even degree. Show that if we remove ANY edge of the graph ...
1
vote
1answer
55 views

Does $z^0$ Have Multiple Solutions?

While playing around with complex numbers, I stumbled upon a result that implies $z^0$ has infinitely many values (where $z$ is any complex number). This struck me as odd since I've never come ...
2
votes
1answer
18 views

Limit of a floor sum

How can i prove that $ \forall x \in \mathbb{R} \displaystyle \lim_{n \to \infty} \dfrac{\left \lfloor{x}\right \rfloor+\left \lfloor{2x}\right \rfloor+\cdots+\left \lfloor{nx}\right \rfloor}{n^2} = ...
2
votes
1answer
36 views

Finding when velocity is zero

Given trajectory $s(t) = 2t^3 - 15t^2 + 36t + 2$ find, when velocity $v = 0$. I'm doing this the following way: $$v = \frac{ds}{dt} = 6t^2 - 30t + 36.$$ Then making $v = 0$, i.e. $6t^2 - 30t + 36 = ...
1
vote
1answer
40 views

Munkres, §22, Exercise 4 (a). Proof verification.

Define an equivalence relation on the plane $X = \mathbb{R}^2$ as follows: \begin{align*} x_0 \times y_0 \sim x_1 \times y_1 \quad \mbox{if} \quad x_0 + y_0^2 = x_1 + y_1^2. \end{align*} Let $X^*$ ...
0
votes
1answer
14 views

Need some clarification of how to add these probabilities.

From a moment generating function I deduced that $P(X=0) = 0.4$, $P(X=1) = 0.3$ and $P(X=3)=0.3$. I am asked to find $P(X>0)$. That would be: $$P(X>0) = 0.3+0.3$$ right?
2
votes
1answer
31 views

Proof Using a Countable Union of Nonempty Open Sets to Prove that $\mathbb{Q}$ Is Infinite

I was playing around with an exercise which shows that if $A \neq \varnothing \subseteq \mathbb{R}$ is an open set, then $A \cap \mathbb{Q} \neq \varnothing$. I then extended the problem to a ...
1
vote
1answer
12 views

Expectation of number of trails till $r$ successes.

Let $X$ be the number of Bernoulli trails till $r$ successes with probability $p$ (including the last one). Find $\mathbb{E}(X)$. My attempt: $$P(X<r) = 0$$ $$P(X=r) = p^r$$ $$P(X=r+s) = ...
1
vote
1answer
42 views

Topology - $f$ is continuous iff $f$ is constant

Let $X_1$ be with the trivial topology, $X_2$ be Hausdorff, $f:X_1 \rightarrow X_2$ a function. Then $f$ is continuous $\iff f$ is constant. I'm not sure that my proof is correct so would appreciate a ...
2
votes
2answers
43 views

Convergence and value of improper integral

I have to prove that integral $I = \int_{0}^{+\infty}\sin(t^2)dt$ is convergent. Could you tell me if it's ok? Let $t^2=u$ then $dt=\frac{du}{2\sqrt{u}}$ Now $$I = ...
1
vote
3answers
50 views

Is this proof for “If $2x+y$ is odd then x is odd or y is odd.” sufficient.

My proof: Consider the contrapositive: "If x and y and both even, then 2x+y is even. 2x+y = even*even + even = even. The contrapositive is logically equivalent, hence the statement is true.
1
vote
1answer
29 views

Let $G$ be a $k$-regular bipartite graph, $k \ge 2$. Prove that every edge of $G$ appears in some perfect matcing in $G$. Is this proof correct?

Using Hall's Theorem, there could only be a perfect matching when the set of $|A|$ vertices have the same number of vertices as set $|B|$, and that there is a subset of vertices $|U|$ in $A$ that is ...
1
vote
0answers
45 views

$\int_{-\infty}^\infty \frac{dz}{z - z_0}$ by contour integration

Consider the integral $\int_{-\infty}^\infty \frac{dz}{z - z_0}$. It has a simple pole at $z = z_0$. Assume $\Im (z_0) < 0$ so the pole is in lower half-plane. Divide $$ \oint_{C_0} = \int_{-R}^R ...
0
votes
0answers
15 views

Doing a project for a CALC II class, and need help determining if we have sufficiently solved a product problem.

We finished our final in Calc II, and now we are doing math projects to pass time until the semester ends. https://docs.google.com/presentation/d/1bYMKfCqcc9zG32Zs_Wti2Fp9rXV7Nbt4QxR3Vnp33Dc As you ...
0
votes
0answers
32 views

Is the composition of uniformly distributed functions uniformly distributed?

Let $\mathcal{I}:=[0,1]$. Def: A measurable function $\varphi:\mathcal{I}\rightarrow \mathcal{I}$ is said to be uniformly distributed with respect to the Lebesgue measure $\Lambda$ if, for any ...
0
votes
1answer
48 views

Is the Riemann zeta function irrational for all integers $n\geq 2$?

The Riemann zeta function for any integer $n\geq 2$ is defined as $\zeta(n) = \dfrac{2^n}{2^n -1}\prod_{p\geq 3} \dfrac{p^n}{p^n -1} $ Observe that $\zeta(n)$ can only be rational if there is ...
2
votes
1answer
27 views

The composition of measurable function is not measurable: only for Lebesgue-measurability?

Let $\mathcal{I}:=[0,1]$. Let $\mathcal{R}(f)$ denote the range of a function $f$. Let $\Sigma$ be the $\sigma$-algebra of $\mathcal{I}$. Consider the measurable and continuous functions ...
0
votes
1answer
35 views

How to fill in the gaps in my proof to make it more convincing?

Let $T$ be a tree with $3$ edges. Let $G$ be a simple graph such that each vertex has degree at least $3$. Show that $G$ has $T$ as a subgraph. This statement is obvious but I am not sure how to ...
0
votes
2answers
40 views

Convergence of $\sum \sin\frac{(-1)^n}{n^p}$

$$\sum_{n=1}^{\infty} \sin\frac{(-1)^n}{n^p}\quad p>1$$ My attempt: $$\sum_{n=1}^{\infty} \sin\frac{(-1)^n}{n^p} = \sum_{n=1}^{\infty} (-1)^n\sin\frac{1}{n^p} $$ And $\sum_{n=1}^{\infty} ...
2
votes
2answers
19 views

Check convergence and sum of a sum of finite sum.

$$\sum_{n=1}^\infty \sum_{k=1}^m \left(\frac{x_k}{y}\right)^n\quad 0<x_k<y$$ My attempt: Convergence: Since $\frac{x_k}{y} <1$ we can conclude that: $$\sum_{k=1}^m ...
4
votes
1answer
55 views

A number $a$ is a square in $\mathbf{Q}$ if and only if it is a square in $\mathbf{R}$ and $\mathbf{Q}_p$ for all primes $p$

Problem from Schikhof's Ultrametric Calculus. As I understand it, the intersection of $\mathbf{R}$ and all $\mathbf{Q}_p$ is just $\mathbf{Q},$ so it seems that $x^2-a$ having a zero in $\mathbf{Q}$ ...
3
votes
0answers
56 views
+50

Is right this application of Hadamard three-lines theorem for $ \frac{\zeta(s)}{s}- \frac{d\zeta(s)}{d\sigma}$?

Let the complex variable $s=\sigma+it$, then from the following identity valid for $\sigma=\Re s>1$ $$\zeta(s)=s\int_1^\infty \frac{[x]}{x^{s+1}}dx$$ where $\zeta(s)$ is the Riemann Zeta function, ...
6
votes
0answers
55 views

If the difference of two independent random variables has a mean, so does each variable

This is a proof-verification request; I’m also recording this proof for my own later reference. Any feedback is appreciated. Claim: Let $X$ and $Y$ be independent, real-valued random variables on ...