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

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0
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1answer
19 views

Prove $f$ is Lipschitz on $K$

Let $f:\mathbb{R}^d\to \mathbb{R}$ such that it's partial derivatives are continuous. Let $K\subseteq \mathbb{R}^d$, a bounded set. Prove that $f$ is Lipschitz on $K$. My work: Since $f$'s ...
0
votes
1answer
7 views

Associated prime of $M/Q$ where $Q$ is $\mathfrak{p}$-primary

I need check if my statement is true and proof check (for some reason I couldn't find this anywhere): Let $Q$ be $\mathfrak{p}$-primary submodule of $A$-module $M$. Then $\mathfrak{p}$ is the ...
0
votes
1answer
44 views

Prove the limit is $e^\alpha$

prove that $\lim_{n \to \infty} \left(1+{\alpha\over n}\right)^n=e^\alpha$ $$\lim_{n \to \infty} \left(1+{\alpha\over n}\right)^n=\lim_{n \to \infty} \left(\left(1+{\alpha\over n}\right)^{n\cdot ...
2
votes
1answer
55 views

Proving a trigonometric identity with tangents [on hold]

Prove that: $$\tan^227^\circ +2 \tan27^\circ \tan36^\circ=1$$ any help, I appreciate it.
2
votes
2answers
22 views

Commutator ideal of reductive Lie algebra

I'm working through Fulton and Harris's book on Representation theory, and I've just done the exercise where I had to show: If $\mathfrak{g}$ is a reductive Lie algebra (defined as $Z(\mathfrak{g}) = ...
0
votes
3answers
58 views

Implies in a truth table, unclear. [duplicate]

In my textbook, we have the following truth table: $P$ true and $Q$ true means that "$P \implies Q$" is true. $P$ true and $Q$ false means that "$P \implies Q$" is false. $P$ false and $Q$ true ...
1
vote
1answer
24 views

Find the Fourier coefficients of $g(x)$

Let $f:\mathbb{R}\to\mathbb{C}$, $2\pi$ periodic function and $f\in C^1$, such that the n-th Fourier coefficient is: $\hat{f}(n) = 3^{-n^2}$. Find the Fourier coefficients of $g(x) = \pi ...
-6
votes
0answers
33 views

Prove by induction this notation [on hold]

Prove by induction? For $n\geq0$, $$\sum_{i = 0}^n (n i+2)^2={1 \over 3}(n+1)(2n+1)(2n+3)$$ Please help me.
1
vote
0answers
32 views

Verification of Basic Proof in Spivak Calculus (Induction)

I have began working through Spivak's Calculus book and trying to do the problems at the end of the chapters. I am rather new to proof, so forgive the naivety of this type of question. I am wanting ...
1
vote
0answers
39 views

Proof of Supporting Hyperplane Theorem from basic definitions.

My purposes in posting this question are twofold. First, I would like to have a lemma which I have proven on the way to proving the Supporting Hyperplane Theorem checked for rigor (zero tolerance for ...
0
votes
0answers
16 views

To circumscribe a square about a given circle.

http://aleph0.clarku.edu/~djoyce/java/elements/bookIV/propIV7.html I was just wondering something . We know that if a line touches a circle at one point, then this means that this line is forming a ...
0
votes
1answer
22 views

How many coins do we need to get $k$ amount

In the far away land of coinsville, they use $4$ different coins as currency, $\{1,10,100,200\}$ What is the computational class of the amount of coins (minimal!!) we need to get $k$ amount? Well, ...
3
votes
3answers
56 views

Strange integral test for convergence in my Analysis Script (proof flawed ?)

Today I was going through my Analysis Script which my Professor used for his course (meaning he often refers to it) and I found a Lemma called Integralcriteria for convergence of Series. I read its ...
2
votes
1answer
30 views

Please check my proof of this elementary covector result

I would appreciate it if someone could look over my proof and verify that it's correct. The question: Let $f$ be a $k$-covector on vector space V. Let $u_1,\dots u_k\in V$ and $v_1,\dots,v_k\in V$ ...
7
votes
5answers
928 views

Proof of the derivative of ln(x)

I'm trying to prove that $\frac{\mathrm{d} }{\mathrm{d} x}\ln x = \frac{1}{x}$. Here's what I've got so far: $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} x}\ln x &= \lim_{h\to0} \frac{\ln(x + h) ...
1
vote
2answers
24 views

Prove that $H=V$ if $H$ is an $n$-dimensional subspace of an $n$-dimensional vector space $V$.

Prove that $H=V$ if $H$ is an $n$-dimensional subspace of an $n$-dimensional vector space $V$. I am not exactly sure what to do to show that $H=V$. So far I have reasoned that since $H$ and $V$ ...
2
votes
4answers
220 views

How do I know which of these are mathematical statements?

While reading this book called "How to Read and do Proofs" by Daniel Solow(Google) I found the following exercise at the end of the first chapter. So how do I know if something is a mathematical ...
1
vote
1answer
36 views

Let $S$ be a subset of an $n$-dimensional vector space $V$, and suppose $S$ contains fewer than $n$ vectors. Explain why $S$ cannot span $V$

Let $S$ be a subset of an $n$-dimensional vector space $V$, and suppose $S$ contains fewer than $n$ vectors. Explain why $S$ cannot span $V$ Proof: Suppose $S$ is a subset of an $n$-dimensional ...
3
votes
2answers
135 views

Show that $f(x) = x^2$ is not uniformly continuous on $[0,\infty)$

Ok, I know the same question has already been asked here, and I am not looking for an answer even though my proof looks kind of the same. But, I need to know whether or not I am on the right track. ...
1
vote
2answers
86 views

if integral $f(x)\cdot g(x)=0$ mean that $f(x)=0$?

The question: If $f(x)$ is a continuous function, such that for every continuous function $g(x)$ defined over $[a,b]$ $$\int_a^b f(x)\cdot g(x)\,dx =0$$ does it mean that $f\equiv 0$? The ...
0
votes
1answer
30 views

The image of the inverse of a continuous function

First of all I'm not sure if my title is correct with the question, I find it hard to really get about what kind of set this question is about. It would be very helpful if someone could explain this ...
0
votes
0answers
19 views

finding the supremum

Let $A=\{x:\frac{[b\cdot n]}{n}\}$ when $n\in \mathbb{N}$ find the supremum we know that $b\cdot n-1<[b\cdot n]<b \cdot n$ therefore $b- \frac{1}{n}<\frac{[b\cdot n]}{n}<b$ so b is a ...
0
votes
1answer
23 views

finding infimum

find the infimum and supremum of $E=\{x \in \mathbb{R}:x=\frac{2}{n}+(-1)^n, n\in \mathbb{N}\} $ $Max(E)=2$ therefore it is also the $Sup(E)$ Let assume that there is $-1<m: m\in E$ so $-1$ ...
1
vote
2answers
51 views

Why $ax^2+bx+c = a(x-r)(x-s)$, where $r$, $s$ are the roots?

When I was reading about math, I came across the following - Suppose the roots of the quadratic $ax^2+bx+c$ are $r$ and $s$. Then $ax^2+bx+c = a(x-r)(x-s)$ for all values of $x$. Is there ...
0
votes
1answer
41 views

Biggest n that can be solved in one second

I have been given the following problem: What is the largest n for which one can solve in one second a problem that requires $(\log_2(n))^2$ elementary operations, where each elementary operation is ...
0
votes
0answers
19 views

How is floor(x) + j/A between x and y, and how the proof is related to the well-ordering property?

I am having trouble with a proof by well-ordering property exercise. Use the well-ordering principle to show that if x and y are real numbers with $x \lt y$, then there is a rational number r with $x ...
0
votes
1answer
33 views

Looking for Clarification on a proof of Density of Q in R

I am looking for some advice/help in regard to the proof that Q is dense in R, given in Walter Rudin's book "Principles of Mathematical Analysis". Mostly, I want to see if my reasoning is correct for ...
2
votes
1answer
65 views

Is this simple calculus proof formal enough and correct?

There is function $f$ differentiable at $x=0$ and $f'(0) = m > 0, f(0) = 0.$ I need to prove that there is $K > 0$ and $\delta > 0$ that for every $0<x<\delta$ : $f(x) > Kx$. So I ...
1
vote
1answer
17 views

An affine set $C$ contains every affine combinations of its points

Show that an affine set $C$ contains every affine combinations of its points. Proof by induction: From the definition of an affine set, we know that $\forall x_1,x_2\in C \text{ and } \theta_i\in ...
0
votes
1answer
14 views

If$\mu$ is $\sigma$- finite, $\epsilon>0$, there exists $A\in \mathcal{A}$ such that $\mu(A)<\infty$ and $\epsilon+\int_A f>\int f$

Problem Let $X\mathcal{A},\mu$ be a $\sigma$-finite measure space. Suppose $f$ is non-negative and integrable. Prove that if $\epsilon>0$, there exists $A\in \mathcal{A}$ such that ...
7
votes
2answers
130 views

proof for $\frac{1}{i} = -i$?

My physical chemistry textbook seems to be making the implicit assumption that $\cfrac{1}{i} = -i$. I'm not sure how this is valid. Here is the snippet of relevant steps: ...
4
votes
3answers
68 views

Show that $f$ is continuous mathematically.

Let $f:[0,\infty)\to \mathbb{R}$ be given by $f(x)=\sqrt{x}$. Show that it is continuous. This is taken from Example 3.7 on <link> page 22 on the paper. It has shown that it is continuous at ...
0
votes
1answer
32 views

Quick question about basic proofs (Spivak)

I was taking a look at some of the practice question from the first chapter of Spivak, and I am wanting to just verify if I am on the right track with things. I do not have solutions, so I am just ...
11
votes
2answers
105 views

Proof that $\sqrt{F!-1}$ is irrational

Please tell me whether my proof is valid. (1) Suppose $\sqrt{F!-1}= \frac p q$ where $p, q$ are integers $>0$ with no common factors. (If there are any common factors we cancel them in the ...
2
votes
1answer
45 views

Analysis: Is $A$ dense in $[0,1]$?

Let $f_n:[0,1]\to\mathbb R$ define by $f_n(x)=\cos(nx)$. Let $A_n=\{x\mid f_n(x)=0\}$ and $A=\bigcup_{n\in\mathbb N}A_n$. I have shown that $|A|=+\infty$. Do you think that $A$ is dense in $[0,1]$ ? I ...
0
votes
1answer
35 views

Point-Set Topology Proof Verification

I am self-studying Baby Rudin (after some idleness) and his second chapter is on basic point-set topology. Now, he asks the following: Is every point of every open set $E\subset\mathbb{R}^2$ a limit ...
2
votes
0answers
17 views

Using Erdős–Szekeres theorem for graph with 50 vertices

Let $G$ be a graph with $50$ vertices such that for every $4$ vertices there are $2$ that have no edge between them (independent). I want to prove that $G$ has a group of $5$ independent vertices.My ...
1
vote
0answers
15 views

Quickest way to restrict a homeomorphism

Let $\phi: U \to V \subset \mathbb{R}^n$ homeomorphism. My desire is: I want to say the restriction $\phi|_{\phi^{-1}(B_{r'}(x))}:\phi^{-1}(B_{r'}(x)) \to B_{r'}(x) $ is a homeomorphism in the ...
2
votes
1answer
52 views

Anywhere I integrate $f_n$, the integral approaches $f$. Is $\lim_n f_n = f$ a.e.?

Something tells me this is obvious... I have a bunch of functions: $f,f_n:\mathbb{R}^2\rightarrow \mathbb{R}$, all integrable. Also, $f$ is continuous. I also have a family of sets, $\mathcal{G}$ ...
2
votes
1answer
43 views

Every continuous map of a closed interval into itself has a fixed point

The Question: Please show this theorem: Let $f: I=[a,b] \rightarrow \mathbb{R}$ be a continuous map such that $f(I) \supset I $. Then $f$ has a fixed point on I. My Attempt: Suppose there is a ...
3
votes
0answers
53 views

Homework problem about the smallest sigma algebra:

Let $\mathscr{E} \subset 2^X $, then there is a unique smallest $\sigma$-algebra containing $\mathscr{E} $. Proof(Attempt): Since $\mathscr{E} \subset 2^X $, $2^X$ is $\sigma$-algebra containing ...
5
votes
1answer
109 views

Question about proving basic results of numbers

I have just recently started to work with Calculus by Spivak and I am just wondering some things about the first chapter. ( I am doing this as a method to review my calculus which I have done but only ...
1
vote
1answer
41 views

$f$ is continuous $\iff f(\bar A) \subset \overline{f(A)}$

The problem is: $f:X\to Y$: any map. $f$ is continuous $\iff \forall A\subset X, \ f(\bar A) \subset \overline{f(A)}$ My understanding is: Suppose $f$ is continuous. $\forall A\subset X, A ...
1
vote
1answer
50 views

Is my approach right? Or is there a better way?

Let $E,F$ normed linear spaces, let $C$ connected of $E$, $D\subset F$, and $f:C\to D$ such that $f$ is open (i.e. sends open sets in $C$ "which is the same as open sets of $E$ intersected with ...
3
votes
2answers
91 views

Proof of the derivative of $x^n$

I am proving $(x^n)'=nx^{n-1}$ by the definition of the derivative: \begin{align} (x^n)'&=\lim_{h \to 0} {(x+h)^n-x^n\over h}\\ &=\lim_{h \to 0} {x^n+nx^{n-1}h+{n(n-1)\over ...
1
vote
1answer
39 views

proof of rational numbers as repeating or terminating decimald

As an exercise in my conceptual algebra class we attempted to determine the reason why this theorem holds true in the forward direction. (Note we decided not to tackle the opposite direction) I wrote ...
1
vote
2answers
78 views

Showing that $f$ is $C^\infty$

Question: Let $f: U \to \mathbb R$ be a continuous function, with $U \subset \mathbb R^2$ open, such that $$(x^2 +y^4)f(x,y) + f(x,y)^3 = 1,\, \,\, \forall (x,y) \in U$$ Show that $f$ is of ...
0
votes
1answer
20 views

Is this statement equivalent to $f(x)\in\mathscr C(a,b)$?

I'm pondering on the following: $$f(x)\in\mathscr C(a,b)\overset{?}{\Longleftrightarrow} f(x)\in\mathscr C[a+\delta,b-\delta]\quad\forall\delta\in(0,\frac12(b-a)) $$ I believe it's true. The ...
1
vote
1answer
62 views
+50

About the usage of the strong approximation theorem

I'm reading Henning Stichtenoth's Algebraic Function Fields and Codes and at Proposition 3.2.5(a) of section 2, chapter 3 he says: Let $\mathcal{O}_S$ be a holomorphy ring of $F/K$. Then $F$ is ...
2
votes
1answer
16 views

Show that $\text{ord}_n (ab) \mid \text{ord}_n (a) \cdot \text{ord}_n (b)$ when $(a,n)=(b,n)=1$ and $(\text{ord}_n (a), \text{ord}_n (b)) = 1$.

The question is to show that if $n$ is a positive integer with $(a,n)=(b,n)=1$ and $(\text{ord}_n (a), \text{ord}_n (b)) = 1$, then $\text{ord}_n (ab) \mid \text{ord}_n (a) \cdot \text{ord}_n (b)$. I ...