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

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On finding the complex number satisfying the given conditions

Question:- Find all the complex numbers $z$ for which $\arg\left(\dfrac{3z-6-3i}{2z-8-6i}\right)=\dfrac{\pi}{4}$ and $|z-3+i|=3$ My solution:- $\begin{equation} \arg\left(\dfrac{3z-6-3i}{2z-8-6i}\...
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1answer
36 views

Proof of Ultrafilter lemma with two propositions and Zorn lemma

I would like to prove the following: Let $X$ be any set, then every filter $\mathcal{F}$ on $X$ is contained in an ultrafilter $F$ Using two propositions and Zorn Lemma. I am required to come ...
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2answers
97 views

What is the name and proof of the limit of this function?

In a proof I find the following limit being used: $$ \lim\limits_{x \to \infty} (1- \frac{a}{x})^x = e^{(-a)}, $$ where $a$ is a constant. Does this limit have a common name and where can I find a ...
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3answers
36 views

Is this enough to prove that the GCD is larger?

Prove that $(a+b, a-b) \geq (a, b)$ My attempt Let $(a+b, a-b) = d$ and $(a, b) = c$. Since $c \mid a,b$ $c$ is also a factor of $a+b$ and $a-b$. Thus $c \leq d$. Is this enough as a proof? It ...
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4answers
118 views

Proving that ${k+x \choose 2k + 1}=-{k-x \choose 2k + 1} $

I need to prove the following, someone can help me? ${k+x \choose 2k + 1}=-{k-x \choose 2k + 1} $ I tried the following: $\frac{(k+x)!}{(2k + 1)!((k+x)-(2k+1))!} = -1\frac{(k-x)!}{(2k + 1)!((k-x)-(...
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1answer
36 views

Show that the reflection of a disc through the origin is a disc

This is a problem from the book "Basic Mathematics" by S.Lang (p.225, exercise 13b). It is similar to the one in my previous question, with the exception that we're considering a reflection instead of ...
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2answers
24 views

Where's the error in this equation of a plane

I was given the question: Find equation of a plane with $P$$(-4,-4,-2)$ and normal vector $\langle -1,4,1 \rangle$. My final answer was: $$-x+4y+z=-10$$ But the last part is wrong $(-10)$. How is ...
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0answers
14 views

Show that the disc of radius $r$ centered at $A$ is the translation by $A$ of the disc of radius $r$ centered at the origin

This is a problem from the "Basic Mathematics" book by S.Lang (p. 225, exercise 12). My problem: Let $D(r, A)$ denote the disc of radius $r$ centered at $A$. Show that $D(r, A)$ is the translation ...
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33 views

help with solution using mengers theorem

to show: for a $3$ regular graph $G$ we have: edge connectivity $=$ vertex connectivity . attempt: take a minimal seperating vertex set $X$ of $G$ with $|X|=:k$. Then $G \backslash X$ has ...
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2answers
53 views

Show that the fourth power of every odd integer is of the form $16k+1$.

This is what I have so far, I'm not sure my reasoning is correct as I am trying to learn how to construct proofs. I would appreciate any feedback on if I took the right steps. If there is an ...
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2answers
52 views

When is $\overline{K}/K$ a Galois extension of $K$?

When is $\overline{K}/K$ a Galois extension of $K$, where $\overline{K}$ stands for the algebraic closure of $K$? I have the following three extensions: $\overline{\mathbb{Q}}/\mathbb{Q}$,$\...
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3answers
59 views

Let $a,b\in \mathbb{N^*}$. Prove that $\gcd(a+b,\operatorname{lcm}[a,b])=\gcd(a,b)$. Is my proof correct?

I made a proof by contradiction. Suppose $δ=(a+b,\operatorname{lcm}[a,b])$ and let it be that $δ\neq(a,b)$. Then $\exists ε\big(ε=(a,b) \land ε\gt δ \big) \implies ε|a \land ε|b \implies ε|(a+b)$. ...
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1answer
33 views

Question about Brouwer degree under uniform convergence.

I was wondering the following: Say a smooth sequence $u_k$ on a smooth manifold converges uniformly to the limit $u$. Does $u$ preserve the Brouwer degree of the $u_k$'s? I also believe this is an ...
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3answers
88 views

Help with Diophantine equation

Prove that the equation $$x^2 - x + 1 = p(x+y)$$ has integral solutions for infinitely many primes $p$. First, we prove that there is a solution for at least one prime, $p$. Now, $x(x-1) + 1$ is ...
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1answer
47 views

Help in proof: a connected graph is $k$ edge connected iff all blocks are

Attempt: we know that the edge set of $G$ is the union of those of it's blocks (maximal connected subgraphs of $G$ not having a cut vertex), any two of them touching in at most one vertex. If all ...
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2answers
94 views

Proof of $\pi^e$ and $e^\pi$ Being Irrational

By contradiction, if $\pi^e$ were rational, then we could write $\pi^e=\frac{a}{b}$ where $a,b\in\mathbb{I}^+$ and $b\neq0$. So: $$\begin{align} \\ \pi^e&=\frac{a}{b} \\ e\ln(\pi)&=\ln(a)-\...
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1answer
25 views

Proving the Hahn Decompostion Theorem from Folland

The Hahn Decomposition Theorem - If $\nu$ is a signed measure on $(X,M)$, there exists a positive set $P$ and a negative set $N$ for $\nu$ such that $P\cup N = X$ and $P\cap N = \emptyset$. If $P',N'$ ...
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1answer
47 views

For which $p$ is $\frac{1}{x^a+x^b}$ in $\cal{L}^p$?

Let $f(x)=\frac{1}{x^a+x^b}$ with $x,a,b>0$. For which $p\ge1$ is $f$ in $\cal{L}^p(\lambda)$ over the interval $(0,\infty)$? Here $\lambda$ is the one dimensional Lebesgue measure. Attempt: We ...
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1answer
96 views

New Proof of Pythagorean Theorem (using inscribed circle)?

I was solving an easy problem for fun when I stumbled onto this, and was wondering if this was a correct and possibly a new proof of the Pythagorean Theorem. Given right triangle $\triangle ABC$, and ...
0
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1answer
27 views

Proving that a matrix transformation maps the zero vector in $\mathbb{R}^n$ into the zero vector in $\mathbb{R}^m$

Suppose $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is a matrix transformation. I want to show that $$T(\vec{0}) = \vec{0}$$ I was told as a hint that we need to use the linearity conditions to make ...
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2answers
151 views

Why is this easy “proof” of Brouwer's Fixed Point Theorem not correct/common?

Brouwer's Fixed Point Theorem states, essentially, that any continuous function on a closed disc to itself has a fixed point. I am familiar with the proof based on the impossibility of a retraction ...
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2answers
67 views

Where is the fault in my proof?

I had some spare time, so I was just doing random equations, then accidentally came up with a proof that showed that i was -1. I know this is wrong, but I can't find where I went wrong. Could someone ...
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3answers
49 views

Diameter of a compact metric space [closed]

Let $(X,d)$ be a compact metric space, then it's bounded, therefore $\operatorname{diam}(X)$ is finite, then there exists $x_0, x_1 \in X$ such that $\operatorname{diam}(X) = d(x_0, x_1)$. Is this ...
2
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2answers
70 views

When is a fact 'obvious enough' that it does not need proving?

Here is an example to better explain the question: Theorem: If $n$ is any integer, then $3n^3 + n + 5$ is odd Counterexample: $n = 2k + 1$ $3n^3 + n + 5 = 3(2k + 1)^3 + (2k + 1) +...
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1answer
43 views

Lusin's Theorem, Modes of Convergence

Background Information: Theorem 1.18 - If $E\in M_{\mu}$, then \begin{align*} \mu(E) &= \inf\{\mu(U):E\subset U, U \ \text{open}\}\\ &=\sup\{\mu(K):E\subset K, K \ \text{compact}\}\end{...
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2answers
31 views

Polynomial ,divides and Induction Proof?

$\text{The polynomial } x-y \;\text{divides the polynomial}\; x^2-y^2 \text{ and } x^3-y^3 \text{because}\; x^2-y^2 = (x+y)(x-y) \text{ and } x^3-y^3=(x-y)(x^2+xy+y^2.) \; \text{for every natural ...
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0answers
41 views

Intersections of algebraic varieties

I was looking at the question here, and I wasn't sure why it was obvious that $V\left(\sum\limits_\lambda I_{\lambda}\right) \subseteq \bigcap\limits_\lambda V(I_{\lambda}).$ But in typing my follow-...
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1answer
35 views

Some questions on Euler's phi function

I was reading Number Theory by George E. Andrews (Dover 1994,) problem set 6-1, p. 81. (I'm not a student; I just find problems like these entertaining like some people enjoy crosswords or Sudoku.) ...
3
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2answers
118 views

Average distance between two random points on a square with sides of length $1$

"What's the average distance between two random points on a square with sides of length $1$?" Here is an attempt which is wrong but I can't see how exactly. Fix $(x, y) \in [0, 1]^2$ The average ...
5
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4answers
147 views

Why is this proof of the chain rule incorrect?

I saw this proof of the chain rule but it says this is a flawed proof. Why? I guessed the reason it is wrong because you can't substitute $g(x+h)$ and $g(x)$ into in limit.
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Formal proof on continuity of multivariable function.

Let $B(o,r) = ${$(x,y)\in \Bbb R: \left\lVert (x,y) \right\rVert<r$}$ $ , to some r>0 and the norm is an euclidean norm, let $f(x,y)$ -> $L$ as (x,y) -> (0,0), with f : $B(o,r)$ -> $\Bbb R$. ...
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0answers
24 views

Marginal mean of two dimensional Brownian motion.

Let $(B^1,B^2)$ be a two-dimensional Brownian motion. Let $t>s$. Is it true that $$ E[B^1(t) \lvert B^2(t),B^2(s) ] = E[B^1(t) \lvert B^2(t),B^2(t)-B^2(s) ] = E[B^1(t) \lvert B^2(t)] $$ since $(x,...
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2answers
19 views

Is this collection of function uniformly equicontinuous? Hints on the proof.

Let $f_n(x)=\frac{1}{n}\cos(e^{nx})$ for $n\in\mathbb{N}$ be a sequence of functions for $x\in[0,1]$. Is it true that $\{f_n\}$ is a uniformly equicontinuous collection of functions? My attempt so ...
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1answer
32 views

Using induction on modified inequalities.

Here's the original problem: Prove by induction that $\left(\frac{1}{2}\right) \left(\frac{3}{4}\right) \cdots \left(\frac{2n-1}{2n} \right) \leq \frac{1}{\sqrt{n+1}}$ for all $n \in \mathbb{N}$. ...
3
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1answer
88 views

Proving that the limit of an integral of a series exists

The goal is to show that the following limit exists $$\lim_{T\to\infty} \frac{1}{T}\int_{-T}^T f(x)dx$$ where $$f(x)=\sum_{n=1}^\infty \frac{e^{ia_n x}}{n^2}$$ I already showed that $f$ is bounded ...
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2answers
39 views

Closure of Union contains Union of Closures

I'm teaching my self topology using a book I found. This is the second part of a 4 part question. First part is here. I'm trying to prove the following problem from a book I found: Let $X$ be a ...
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1answer
47 views

Function with countably many points of discontinuity

Aside from rigor, is this proof correct? Claim. Let $f$ be a function defined on $[0, 1]$ such that $\lim\limits_{y\to a} f(y)$ exists for all $a \in [0, 1]$. Then for any $\epsilon > 0$ there are ...
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0answers
28 views

Closure of Intersection is Subset of Intersection of Closures

I'm trying to prove the following problem from a book I found: Let $X$ be a topological space and let $\mathscr{A}$ be a collection of subset of $X$. Prove $\overline{ \bigcap \limits_{A \in \...
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0answers
99 views

Find the limit of the mediant sequence $\frac{p_{n+1}}{q_{n+1}}=\frac{ap_n+bk_n}{aq_n+bm_n},~\frac{k_{n+1}}{m_{n+1}}=\frac{cp_n+dk_n}{cq_n+dm_n}$

How to find the general formula for the limit of the sequence: $$\frac{p_{n+1}}{q_{n+1}}=\frac{ap_n+bk_n}{aq_n+bm_n}$$ $$\frac{k_{n+1}}{m_{n+1}}=\frac{cp_n+dk_n}{cq_n+dm_n}$$ $$a,b,c,d>0$$ $$...
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3answers
86 views

Factorial Proof by Induction Question? [duplicate]

$\text{Use the PMI to prove the following for all natural numbers n.}$ $ \frac{1}{2!} + \frac{2}{3!} + \cdot \cdot \cdot + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!} $ So for this question I get ...
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8answers
104 views

Prove: $\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}\geq \sqrt{x}+\sqrt{y}$

Prove: $$\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}\geq \sqrt{x}+\sqrt{y}$$ for all x, y positive $$\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}-\sqrt{x}-\sqrt{y}\geq 0$$ $$\frac{x\sqrt{x}+y\sqrt{y}-x\sqrt{y}-...
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1answer
25 views

Closed disjoint sets in $\omega_1$ implies one of them is countable

Let $A, B \subseteq \omega_1$ be disjoint, closed sets. Show that one of A or B must be countable. Since $\omega_1$ is uncountable, at least one of A or B must be countable, I want to show that ...
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1answer
57 views

Need help in understanding some basic ordinal concepts

I'm trying to construct a proof that there is no homeomorphism from a subspace of $\omega_1$ to $\Bbb{Q}$ with the usual topology, but first I need to clear up some concepts. This is the definition ...
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1answer
52 views

Proof of indexed cartesian product

Could anyone verify if my proof is correct please? The question is as follows: For each $(i,j)\in I\times I$, let $C_{i,j}=A_i\times B_j$, and let $P=I\times I$. Prove $\bigcup_{p\in P}C_p=\...
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1answer
16 views

How to show a set is compact using sequential compactness definition?

Let $l^{\infty}$ be the vector space of all bounded sequences $x=(x_n)$ of real numbers with the norm $||x||=\sup_{n\in\mathbb{N}}|x_n|$ and $l^{\infty}$ is complete. I am trying to show that the set ...
0
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1answer
18 views

Given any $r \in \mathbb Q$, there is $m \in \mathbb Z$ such that $m \le r < m + 1$

Suppose the set $S$ contains all $n \in \mathbb Z$ such that $n > r$ for any $r \in \mathbb Q$. By Archimedes, there are some $m, n \in \mathbb Z$ such that $n > r > m$ for any $r \in \mathbb ...
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0answers
32 views

Prove: $\mid\sum_{i=1}^{n} x_i\mid\leq \sum_{i=1}^{n}\mid x_i\mid$ [duplicate]

$\mid\sum_{i=1}^{n} x_i\mid\leq \sum_{i=1}^{n}\mid x_i\mid$ If $n$ is even we will divide the sum into groups of $2$ $x$'s namely $\mid x+x \mid \leq \mid x\mid+\mid x \mid$ and will repeat the ...
0
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2answers
43 views

There is group a $S$ with $2n$ members $n$ of them are identical and $n$ of them are different, How many subsets are there?

I have the following question : There is a set $S$ with $2n$ members $n$ of them are identical and $n$ of them are different, How many different subsets are there for $S$ in size $n$. This is what I ...
0
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1answer
55 views

Proving that every set has a finite subset of ``smallest elements''.

Update: I would like to have this deleted but I'm behind a firewall which prevents from using most features -- including commenting. Reason for deleting: $c-x\in \Bbb N_0^n\setminus\{(0,0,\dots,0)\}$ ...
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1answer
68 views

Finite Field Question: Which of the followings are true?

I have the following True or False question that I am having trouble getting it correct. I've written down my thoughts on each choice. If anyone could verify my thoughts or tell me where I made a ...