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

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0answers
33 views

Multiplicative implication of Goldbach's conjecture?

I've recently been thinking what would goldbach's conjecture imply in multiplicative expression: We start by stating Goldbach's conjecture in terms of the powers of a polynomial: $$ f(x)^2 = (\sum_{...
-3
votes
0answers
21 views

Let K be an integer between $800,000$ and $900,000$ so that (Greatest Common Divisor) [closed]

Let K be an integer between $800,000$ and $900,000$ so that,$\gcd(K,271)>\gcd(K,2016)>100$. List all values of K. Need serious help with this!!! Respond asap, please!
3
votes
1answer
91 views

Is my proof ok? If $\sum u_n$ diverges then $\sum \frac {u_n} {u_1 + u_2 + \dots + u_n}$ also diverges

The question is : If $\sum u_n$ is a divergent series of positive real numbers and $s_n = u_1 + u_2 + \dots + u_n$ , prove that the series $\sum \frac {u_n} {s_n}$ is divergent. I tried my best. ...
2
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1answer
23 views

Describe all smooth surfaces in $\mathbb{R}^3$ with coordinates $(x,y,z)$ such that the pullback of the one-form $\theta:=dy-zdx$ is identically zero.

My question is as the title states: Describe all smooth surfaces in $\mathbb{R}^3$ with coordinates $(x,y,z)$ such that the pullback of the one-form $\theta:=dy-zdx$ is identically zero. Now, ...
0
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2answers
48 views

Check my work: Evaluating $\tan\frac{7\pi}{8}$ using a half-angle formula

I am doing a trig problem involving half-angle identities, and I am not sure if my solution is correct. Can someone please check my work? The question: Find the exact value of $\tan\frac{7\pi}{8}...
0
votes
0answers
48 views

Calculus, limit at infinity exists, bounded second derivatives

Let $f:[0,\infty) \to \mathbb{R}$ twice differentiable. If $f''$ is bounded and $\lim_{x\to \infty} f(x)$ exists, show that $\lim_{x\to \infty} f'(x) = 0$. Update: So following the link from one of ...
0
votes
3answers
34 views

How to show that any separable space is CCC

I thought I had the proof of this in my head, but it doesn't sound right on paper. Can someone see if my argument could be improved. Let $(X,\tau)$ be a topological space that is separable, then it ...
0
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0answers
15 views

Linearity of projection of angle

In the book Putnam and Beyond, problem 252 reads as follows: Consider the angle formed by two half-lines in three-dimensional space. Prove that the average of the measure of the projection of the ...
6
votes
1answer
53 views

Does there exist a multiplicative $f:\mathbb{Q}^+\to\mathbb{Q}^+$ such that $f\neq x\mapsto x^a$ for all $a$?

If we consider the functional equation: $f:\mathbb{Q}^+\to\mathbb{R}$ such that $$ f(xy)=f(x)f(y) $$ for all $x,y\in\mathbb{Q}^+$ I think, I have constructed a solution which is not of the form $x\...
1
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1answer
21 views

In a metric space $X$, a subset $S \subset X$ is relatively sequentially compact if and only if its closure $\overline{S}$ is sequentially compact.

Prove that in a metric space $X$, a subset $S \subset X$ is relatively sequentially compact if and only if its closure $\overline{S}$ is sequentially compact. The terms relatively sequentially ...
0
votes
1answer
18 views

Discrete Math Proof: Divisibility equivalence

For all integers $a$, $b$, $d$, if $d$ divides $a$, and $d$ divides $b$, then $d$ divides $(3a+2b)$ and $d$ divides $(2a+b)$. Prove the statement. What Assumptions do I need to make at the beginning ...
1
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1answer
35 views

Discrete Math Proof verification: products of floor

Determine if the following is true or false and provide a proof: $\forall x\in\mathbb{R},\exists y\in\mathbb{R}$ so that $\lfloor xy\rfloor = \lfloor x\rfloor \lfloor y\rfloor$ My attempt: -The ...
1
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0answers
34 views

Finite dimensional separable algebra is étale

Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every extension $L/\Bbbk$. Say it's étale if there's an extension $L/\Bbbk$ such that $L\otimes_\Bbbk A\cong \prod_1^nL$. Here'...
2
votes
2answers
86 views

About a proof that $\lfloor x^2\rfloor = \lfloor x\rfloor^2$ for unbounded non integer values of $x$

I am taking a first course in discrete mathematics. The instructor parsed the following question that has the following solution, respectively: Prove the statement: For all positive integers $N$, ...
1
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1answer
44 views

Alternative proof: show that any metrizable space $X$ is normal - Part 1

There is a proof online that shows that all metric spaces are normal. The proof is as follows However, it has the additional baggage of needing to show that $d(x,A)$ is continuous and $U,V$ are ...
1
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1answer
35 views

The set of all real or complex invertible matrices is dense

I'm trying to show that the set of all invertible matrices $\Omega$ is dense over $F=\mathbb R$ or $\mathbb C$. Let $A\in\Omega$ and $C\in M_{n\times n}(F)$. Since $\|A-C\|<\frac{1}{||A^{-1}||}$, ...
1
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2answers
19 views

Show that any metrizable space $X$ is regular

This is a quick follow up to another question Show that any metrizable space $X$ is Hausdorff Recall, a topological space $(X,\mathcal{T})$ is regular if we can separate any point $x$ from ...
0
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0answers
22 views

Prove that if $A_1,A_2,…,A_n$ are finite sets, then $\bigcup_{i=1}^{n} A_{i}$ is finite. [duplicate]

Let, S be a set defined such that $ S =\{n \in \mathbb{N}:$ if $A_1,A_2,...,A_n$ are finite sets, then $\bigcup_{i=1}^{n} A_{i}$ is finite$ \} $. Let n = 1 and assume $A_n = A_1 $ is finite. Then, $\...
1
vote
2answers
46 views

Which of the following subsets of $\Bbb R^2$ are homeomorphic to the set $\{(x, y) \in \Bbb R^2 \mid xy = 1\}$?

Which of the following subsets of $\Bbb R^2$ are homeomorphic to the set $\{(x, y) \in \Bbb R^2 \mid xy = 1\}$? $a. \{(x, y) ∈ \Bbb R^2 \mid xy − 2x − y + 2 = 0\}.$ $b. \{(x, y) ∈ \Bbb R^2 \...
1
vote
1answer
38 views

Poisson Distribution - Optimistaion

A store offers a new seasonal product featured. Let $N$ be the random variable which means the number of clients who come to the store during the season, where $N \sim \operatorname{Poisson}(19)$. ...
1
vote
2answers
81 views

Proving the Baire Category Theorem from scratch, Stuck!

Theorem: (Baire) $(X,d)$ is a complete metric space then the intersection of countably many dense, open subsets in the metric topology $\mathcal{T}$ generated by $d$ is dense In other ...
6
votes
3answers
69 views

Show that linear functional $L(f) = \int_0^1 f(x) dx$ is continuous

Let $(C[0,1], d_1)$ be a metric space of all continuous functions $f:[0,1] \to \mathbb{R}$, $d_1$ is the $L_1$ metric $$d_1(f,g) = \int\limits_0^1 |f(x) - g(x)| dx$$ Show that linear functional $L(...
1
vote
1answer
50 views

Show all sequence of $l^1$ with $|x_n|\leq \frac{1}{n^2}$ is compact.

Could you help me to check my proof: let $\{x^k\}$ be a sequence in such set, we use Cantor's diagonal argument to show the existence of convergent subsequence. There exists a subsequence $\{x^{\...
1
vote
0answers
54 views

Let $u_n > 0 ,v_n > 0$ for all $n$ which are bounded, then $(\lim \sup u_n)\cdot(\lim \sup v_n) \geq \lim \sup u_n v_n$.

The question is : Let $\{u_n\}$ and $\{v_n\}$ be two bounded sequences such that $u_n > 0$ and $v_n > 0$ for all $n \in \mathbb {N}$, then show that $(lim \sup u_n).(lim \sup v_n) \geq lim \...
4
votes
1answer
49 views

Why all vector space have a span set?

I thought about this question, but I don't sure if my proof is correct. In the book, he put this question like a observation of span sets' definition, so I tried proof this. My attempt: Suppose that ...
5
votes
1answer
80 views

Open interval $(0,1)$ with the usual topology admits a metric space

which of the following is/are true ? $(0,1)$ with the usual topology admits a metric which is complete . $(0,1)$ with the usual topology admits a metric which is not complete. $...
1
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1answer
30 views

Show a set $C$ is closed iff point to set distance is not zero for all points outside of the set

I think this is an interesting problem Suppose that we have $X$ with the metric topology $\tau$ such that $C$ is a closed set in $\tau$ Define the point to set distance as $$dist(x, C)...
4
votes
1answer
57 views

Prove that if $A$ has the same cardinality as $\emptyset $, then $A= \emptyset$

Let Equivalence($\equiv$) be defined as having the same cardinality. Assume, $A \equiv \emptyset $. Then, there exists a function $f: A \to \emptyset $ such that f is a bijection. This proof will ...
0
votes
0answers
36 views

Proving that an infinite series equal a finite series

Suppose we have a function $f(z)$, which has $m$ isolated singularities, which are non-integers (say, $z_1$, $z_2$,...,$z_m$). Define $H(z):=\frac{\pi f(z)}{\sin(\pi z)}$. Assume that there exists a ...
1
vote
3answers
58 views

Proving $ \bigcup_{i=1}^n A_{i} \text{ is finite.} $ by Induction.

Prove : If $A_{1},A_{2},...,A_{n} \text{ are finite sets, then } $$$ \bigcup_{i=1}^n A_{i} \text{ is finite.} $$ Proof: (I) Basis Step : $p(1)$ is true because it is true because it is finite. ...
2
votes
0answers
34 views

Show that if $(X, \mathcal{T}), (Y, \mathcal{J})$ are both metrizable, then $X \times Y$ with product topology is metrizable

Let $(X, \mathcal{T}), (Y, \mathcal{J})$ be both metrizable, then Claim: $X \times Y$ with product topology is metrizable I have made an attempt at this question but the notation is ...
2
votes
0answers
45 views

Elementary proof of $C^\infty_c$ is dense in $L^p (L^q)$ mixed space

As it well known, $C^\infty_0 (\mathbb{R}^n)$(the space of infinitely differentiable functions with compact support) is dense in $L^p (\mathbb{R)^n}$ Here I want to consider the same result with ...
0
votes
1answer
66 views

On the inequality $|z_1-z_2|^2 \lt (1+c)|z_1|^2+(1+\frac{1}{c})|z_2|^2$

Now, I know this question has been asked here but my question doesn't deal with finding a solution, my question deals with checking the validity of the question. Question:- If $z_1, z_2$ are ...
1
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1answer
25 views

Proofs involving limsup and liminf

I've been working with proofs involving $\limsup$ and $\liminf$, and I'm a bit confused regarding their general methodology. More specifically, I'm unsure about whether my approach to the following ...
1
vote
1answer
101 views

Prove that $\displaystyle\int_0^\infty \frac{\sin x}{x}dx$ converges using power series

$$\sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}-...=\displaystyle\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)!}$$ $$\frac{sin x}{x} = 1-\frac{x^2}{3!}+\frac{x^4}{5!}-...=\displaystyle\sum_{n=0}^\infty(-1)^...
3
votes
0answers
26 views

$f$ is uniformly continuous only if $g$ is constant

Let $g:\mathbb R\to\mathbb R$ be continuous and define $f:\mathbb R^2\to\mathbb R$ by $f(x_1,x_2)=g(x_1x_2)$. Show that $f$ is uniformly continuous only if $g$ is a constant function. I'm not sure ...
1
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1answer
39 views

Ampleness and Extensions of Line-Bundles

I believe I have a proof that any vector bundle $V$ of rank $n$ on a projective variety $X$ has a filtration by line bundles (that is, there is a filtration $V = L_n \subset ... \subset L_0$, where $...
1
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2answers
26 views

Endomorphisms and nilpotency

I tried to proof something about nilpotent matrices but I'm not quite sure if my proof is correct, it is at least a bit handwavy so I'd love if someone would explain to me how I should make it more ...
0
votes
0answers
28 views

Double Integration in Polar Coordinates

$\iint 2x-y$ $dA$ in the first quadrant and enclosed by $x=0$ $y=x$ and $x^2+y^2=4$ Since the function is enclosed in the first quadrant then $0 \leq \theta \leq \frac{\pi}{2}$ and since $y=x$ and $x=...
0
votes
1answer
47 views

transport functions ,lemmas between isomorphic structures using univalence

According to the HoTT book page 153,we can get functions and lemmas with ease using univalence,for example,in order to get the double'(double function for N') function from double(for N),we can just ...
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10answers
117 views

A clean proof of $x^2 \geq x$, for any integer $x$

I am trying to prove that $x^2 \geq x$ for any integer $x$. Since we know that for any number $n$, $n^2 \geq 0$ we conclude that if $x \leq 0$ the proposition will hold. Next we must prove that the ...
2
votes
2answers
57 views

Does this series converge conditionally $\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{\frac{1}{10}}}$

$\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{\frac{1}{10}}}$ According to my understanding, if $\sum\left|a_n\right|$ diverges but $\sum a_n$ converges, then the series is conditionally convergent. For $\...
4
votes
0answers
28 views

Proof validation: complete set, change of variable

Let $\phi(x) \in \mathcal{C}^1([0,1])$ be a real valued function such that: $$\begin{cases} \phi'(x) > 0 & \forall x \in [0,1] \\ \phi(0) = 0 \\ \phi(1) = 1. \end{cases}$$ I'm asked to prove ...
3
votes
1answer
59 views

Real Analysis, Problem 3.2.14 The Radon Nikodym Theorem

Problem 3.3.14 - If $\nu$ is an arbitrary signed measure and $\mu$ is a $\sigma$-finite measure on $(X,M)$ such that $\nu\ll \mu$, there exists an extended $\mu$-integrable function $f:X\rightarrow [-\...
0
votes
1answer
58 views

Brute force way to show that $\rho(x,y) = \min\{1, d(x,y)\}$ is a metric

Following a hint in Short proof that $\rho^\prime(x,y) = \min\{1,\rho(x,y)\}$ is a metric I would like to use the brute force method to show that the standard bounded metric is a metric $$\rho(x,y) ...
1
vote
1answer
32 views

Double integration over a general region as a type I

$\iint xy$ bound by the curves $y=x^2$ and $y=3x$ The easiest way to integrate this is as a type I integral $dydx$ since $3x \gt x^2$ in the interval $3x$ is my upper bound and my interval is $[0,3]$ ...
2
votes
2answers
69 views

Follow up to a question, why does proof $\rho(x,y) = \dfrac{d(x,y)}{1+d(x,y)}$ work

This is a follow up to a well known question Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric A general proof is as follows: Let $x,y,z \in (X, \rho)$ \begin{align} \rho(x,z) &= \dfrac{...
1
vote
0answers
51 views

Proof Attempt: Non-decreasing continuous CDF is standard uniformly distributed

Proof Attempt: For any random variable $X$ with non-decreasing continuous cdf $F(x)=\Pr(X≤x)$ (note that the inverse function does not necessarily exist due to flat regions in $F$), I wish to prove ...
1
vote
1answer
42 views

Group of order $pq$ with $p<q$ prime has $q-1$ elements of order $q$

If $p<q$ are primes and $G$ is a group of order $pq$, then $G$ has exactly $q-1$ elements of order $q$. My attempt was: Use Sylow theorem to show that $G$ has only one subgroup of order $q$, so ...
1
vote
1answer
56 views

Show that a set with an uncountable subset is itself uncountable.

Let $A = P \cup Q$, where $P, Q$ are disjoint [1] and $P \ne \emptyset$ is countable and $Q \ne \emptyset$ is uncountable. Then $Q \subset A$ [2]. Show that $A$ is uncountable. Proof (by ...