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

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Proof that $\{\,\left]a,\infty\right [\mid a\in\mathbb{R}\,\}\cup\{\mathbb{R} \}\cup\{\emptyset \}$ is topology of $\mathbb{R}$

Proof that $\mathcal{T}:=\{\,\left]a,\infty\right [\mid a\in\mathbb{R}\,\}\cup\{\mathbb{R} \}\cup\{\emptyset \}$ is topology of $\mathbb{R}$. I have slight trouble on writing this down.. I'll first ...
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1answer
29 views

Two questions on Munkres -Topology

I have two questions: If $X$ is a countable product of spaces having countable dense subsets then does $X$ have a countable dense subset? Let $X$ $=\prod_{i=1}^\infty X_i$ .Let $D_i$ denote the ...
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0answers
19 views

Function is differentiable in all the points of its domain

I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function $\in C^k$ and a function is $C^k$ if is composition of ...
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1answer
31 views

Munkres Topology Article -30 Problem 5

Show that a metrizable space with a countable dense set has a countable basis. My try: Let $X$ be a metrizable space with a countable dense set $D$. Consider for each $n\in \Bbb ...
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4answers
136 views

Show that every compact metrizable space has a countable basis

Show that every compact metrizable space has a countable basis. My try: Let $X$ be a compact space and metrizable. Now for each $n\in \Bbb N$; I can consider the open cover $\{B(x,\frac{1}{n}):x\in ...
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1answer
34 views

Completeness of bounded linear maps

Let $X,Y$ be normed vector spaces over $\mathbb{C}$, and $L(X,Y)$ the space of all bounded linear maps from $X$ to $Y$. Its known that $L(X,Y)$ is a normed(operator norm) vector space. Theorem: ...
2
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1answer
35 views

Proving that a complex-valued function has limit infinity at a finite point

Question: Show using the $\epsilon -\delta$ definition that $$\lim \limits_{z \to i} \frac{z-1}{z^2+1} = \infty$$
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3answers
40 views

Simplify $(k +1)! > (k + 1)^2$ to prove true for $k ≥ 4$

I am trying to prove this statement is true for $k ≥ 4$. I don't know how to work with $k + 1$ factorial, so I'm a little lost on proving this.
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3answers
52 views

Can I further simplify $5^k \cdot 5 + 9 < 6^k \cdot 6$ to prove this is true

I am trying to prove this statement, but I'm not sure where to go from here. Is don't think this is sufficiently reduced to conclude the statement is true, but I'm not positive. $k ≥ 2$ Can I ...
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3answers
47 views

Prove $n^3 + 7n + 3$ is divisible by 3 for all integers n ≥ 0

The statement I'm trying to prove is: $n^3 + 7n + 3$ is divisible by 3 for all integers n ≥ 0 I eventually need to prove $(k + 1)^3 + 7(k + 1) + 3$ is divisible by 3. I don't really understand ...
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2answers
25 views

Help Proving the Average is greater than B^(1/n)

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Define the following two numbers: $A = (a_1 + a_2 + \cdots + a_n) /n $ (The average of the numbers) $B = (a_1 + a_2 + \cdots + ...
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1answer
46 views

Show $ (\int_{-\infty}^\infty \sqrt{p}\sqrt{q}d\mu)^2\leq 2 \int_{-\infty}^\infty \min\{p,q\}d\mu $

Consider a random variable $X$ in $(\Omega, \mathcal{F}, \mathbb{P})$. Let $p,q$ be two densities with respect to a measure $\mu$ in $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where ...
0
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1answer
18 views

Continuity proving of function with delta-epsilon

Prove continuity of function with the delta-epsilon definition in point $x_o=0$ $$f:\mathbb{R}\rightarrow \mathbb{R}$$ $$f(x) = \begin{cases} x^2+1, & x \in \mathbb{Q} \\[2ex] 2^x, & x \in ...
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0answers
16 views

Lebesgue measure of region under curve

Let $(X,\Sigma,\mu)$ be a $\sigma$-finte measure space and $f \in L^+(X,\Sigma)$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$. Theorem: Define the area under the graph of $f$ to be ...
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1answer
34 views

if $r,s$ are rational numbers, then $r+s\sqrt2$ is irrational unless $s=0$?

if $r,s$ are rational numbers, Prove $r+s\sqrt2$ is irrational unless $s=0$? I need to prove this simple question, but not sure if my method is acceptable I'm trying to prove it by ...
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1answer
13 views

Prove that: $n^2+3n^3 + 6^{lgn} is $ $\theta(n^3)$

I'm asked to prove that: $n^2+3n^3 + 6^{lgn} is $ $\theta(n^3)$ I know that for Big O, I need to show: $f(n) <= c*g(n)$ But I'm not sure how to show this, since it involves theta. Any help would ...
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2answers
49 views

Prove for all integers n such that n ≥ 3, $ 4^3 + 4^4 + 4^5 … 4^n = \frac{4(4^n - 16)}{3}$

I am trying to prove this using mathematical induction, but I'm lost once I get to comparing the two sides of the equation. Proposition: For all integers n such that n ≥ 3, $ 4^3 + 4^4 + 4^5 … ...
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0answers
20 views

Riemann-Stieltjes Integral Substitution

I want to prove $\int^b_a\,f(g(x))\,dg(x) = \int^{g(b)}_{g(a)}\,f(x)\,dx$ for all f continuous. Firstly, $\int^b_a\,f(g(x))\,dg(x) = \int^b_a\,f(g(x))g'(x)\,dx$, since g is continuous and ...
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1answer
20 views

Do these non-homotopic maps induce the same map in reduced homology?

Consider two maps $f, g: X\to Y$, where $X=Y=\{ 0, 1 \}$ with discrete topology, $f$ is the identity and $g$ maps everything to 0. Then it's clear that $\widetilde{H}_0(X;\mathbb{Z})\cong \mathbb{Z}$ ...
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0answers
17 views

Verifying a startegy to prove convexity on partial domain

Assume you have the multivariate function $$f(x_1,x_2,..,x_n)$$ where: $x_i>0 \forall i$, and $\sum_i x_i = 1$. I need to show that $f$ is a convex function. My plan is to show that it is ...
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0answers
12 views

If $u \in W^{1,p}(I) \bigcap C_c(I)$ then $u \in W_{0}^{1,p}(I) $

I want to show the following statement: If $u \in W^{1,p}(I) \bigcap C_c(I)$ then $u \in W_{0}^{1,p}(I) $. $W^{1,p}(I) $ is the Sobolev Space, i.e. the space consisting of the functions that ...
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2answers
40 views

Find the polynomial $P$ of smallest degree with rational coefficients and leading coefficient $1$ such that $ P(49^{1/3}+7^{1/3})=4 $

Find the polynomial $P$ of smallest degree with rational coefficients and leading coefficient $1$ such that $$ P(49^{1/3}+7^{1/3})=4 $$ (Source:NYSML) My attempt Let $$ ...
2
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1answer
14 views

Loss of Dimension with Orthogonal Projection

I am working on trying to prove this claim: Consider a nonzero vector v in Rn. What is the dimension of the space of all vectors in Rn that are perpendicular to v? I think I intuitively understand ...
2
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2answers
28 views

Solvability of nilpotent groups

I'm uncertain about my proof about this exercise regarding nilpotent groups. If someone could me help me out, that would be appreciated. There's a post about this problem, but it uses another ...
0
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0answers
58 views

The condition that given polynomial is divisible by 3

In How can I prove that the following is divisible by 3?, I showed $k^3+3k^2+2k$ is divisible by $3$ using Euler's theorem, specifically, Fermat's little theorem. Then I thought that it is possible to ...
0
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1answer
24 views

For a sequence, why must $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {||x_n||} = 0$, or there exists a convergent subsequence with a nonzero limit?

Suppose I've got a sequence of vectors $\{x_n\}_{n∈N}$ in $\mathbb{R}^k$. Why is it that exactly one of the following three facts must hold: $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {x_n} = 0$, or ...
1
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2answers
42 views

If $(d,a)=1$ and $d|ab$ then $d|b$ .

Okay, checking to see if i'm on the right track. I essentially did the same prove for Euclid's lemma but exchanged the $d$ for the $p$. Is that the right idea? Or am I missing something?
0
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1answer
27 views

For the same Conditional Probability, why does Bayes's Theorem differ from a direct calculation?

Abbreviate: S = a person is diseased, + = the test is positive. Presume: $\Pr(D) = 0.001, \; \Pr(+|D)=0.99, \; \Pr(+|D^C) = 0.01 \qquad ($$\iff$ $ \Pr(-|D^C) = 0.99).$ 1. Use Bayes's Theorem: ...
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0answers
25 views

Well-ordering principle proof via Analysis

I would appreciate if my proof attempt could be evaluated, and some hints could be given. I think that, perhaps, my proof is not ideal. Prove: If $E$ is a non-empty subset of $\mathbb{N}$ then $E$ ...
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2answers
13 views

Decreasing sequence and prove by contradiction

I have "solved" the following question using prove by contradiction. But it seems a bit off to me: Let {$x_k$} be a sequence satisfying $x_{k+1}\le(1-\beta)x_k$ for $0\lt\beta\lt 1$ , and $x_0\le C$. ...
0
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1answer
17 views

An exercise about basis for orthogonal subspace (solution check)

I believe what I did in this exercise is correct, but I'm wondering if there is a faster way to do this kind of computation. I'm practicing for an exam that requires me to be really fast solving ...
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2answers
50 views

Proving continuity with epsilon delta

I have the function $f:\mathbb{R}\rightarrow \mathbb{R}\:\:f\left(x\right)=x^2-3x$ and it asks me to prove continuity in point $\:x_o=0$ using the epsilon-delta definition. I know that in order to do ...
0
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1answer
38 views

Prove that $\lim_n (-n+\sin(n) )= -\infty$

Prove that $\lim_n (-n+\sin(n)) = -\infty$ So I need to show that if $B'$ is any number, then there is a number N' such that $$n>N' \implies S_n\lt B'$$ I am having trouble feeling confident in ...
0
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1answer
31 views

Prob. 9, Sec. 19 in Munkres' TOPOLOGY, 2nd edition: Equivalence of the choice axiom and non-emptyness of Cartesian product

The Axiom of Choice is as follows: Given a collection $\mathcal{A}$ of disjoint non-empty sets, there exists a set $C$ consisting of exactly one element from each element of $\mathcal{A}$; that ...
3
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0answers
52 views

Prove an annulus is homeomorphic to a cylinder

Let $A \subset \mathbb{R}^{2}$ be the annulus $A = \{(x,y) \in \mathbb{R}^{2} \colon 1 \leq x^{2} + y^{2} \leq 4 \}$. Prove that $A$ is homeomorphic to $S^{1} \times I$, where $I = [0,1]$ is the ...
0
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1answer
46 views

Do we have $\frac{1}{a} - \frac{1}{b} = b - a$?

I am attempting to prove that $$\frac{1}{E'} - \frac{1}{E} = \frac{1}{m_e c^2} \cdot (1-\cos\theta)$$ can be derived from $$E + m_ec^2 - E' = c^2(p^2 - 2pp'\cos\theta + p'^2) + m_e^2c^4 $$ where ...
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2answers
35 views

If $A$ and $B$ are conneted and $A\cap B\neq \emptyset$, then $A\cup B$ is connected

Can you please let me know if my proof is reasonable? Prove: If $A$ and $B$ are conneted in $\mathbb{R}^n$ and $A\cap B\neq \emptyset$, then $A\cup B$ is connected Proof: Suppose that $A\cap B$ is ...
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4answers
54 views

Prove that if $n$ is odd, then $-n$ is odd.

Here is my work so far, I am missing something quite obvious but I can't seem to link it together: Proof. Let $n$ be an integer. Suppose $n$ is odd. This means that there is an integer $k$ such that ...
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1answer
24 views

Prove that $\sum_{1\le k \le x}a_kf(k)=f(x)\sum_{1\le k \le x}a_k-\int_1^xf'(t)\left( \sum_{1\le k \le t} a_k\right)\,dt,\,\,\,\,\,x\ge 1$

If $a_1, a_2, \dots \in \mathbb{R}$ and $f$ is a $C^1$ function in an open set that contains $[1, \infty)$, prove that $\sum_{1\le k \le x}a_kf(k)=f(x)\sum_{1\le k \le x}a_k-\int_1^xf'(t)\left( ...
0
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0answers
15 views

Is my proof correct? Convex optimization

There's a theorem that says that if $C \subset \mathbb R^n$ is a convex set, then $x^* \in C$ is the closest point in $C$ to $y \notin C$ if and only if $(y-x^*)\cdot(x-x^*)\leq 0$ for all $x \in C$. ...
0
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1answer
20 views

Calculate the vector surface integral

Let $V=\{(x,y,z)\in \mathbb{R}:\frac{1}{4}\le x^2+y^2+z^2\le1\}$ and $\vec{F}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{(x^2+y^2+z^2)^2}$ for $(x,y,z)\in V$. Let $\hat{n}$ denote the outward unit normal ...
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0answers
37 views

Wrong result: a continuous function has zero $p$-variation, for every $p$. Where's the error?

Let $\Pi_n$ be a sequence of partitions with $|\Pi_n| \to 0$. Then the $p$-variation of a continuous function $g$ along the partitions $\Pi_n$ is defined as $$V_T^p(g) = \lim_{n \to \infty} V_T^p(g, ...
2
votes
2answers
53 views

Proving volume of a sphere

I randomly decided to derive the volume of a sphere. The area of a circle is $\pi r^2$. So the volume, I thought, should be $\int \pi r^2 dr = \frac{\pi r^3}{3} $, summing up the area of many discs. ...
21
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2answers
506 views

Would this solution of the limit of the sequence be correct?

Let's suppose that I have the sequence $a_n = \frac{1}{n^2} + \frac{2}{n^2} + \frac{3}{n^2} + \ldots + \frac{n}{n^2}, n \in \mathbb{N}$. And I have to find the limit of the sequence as $n \rightarrow ...
1
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1answer
46 views

$\mathbb CP^1 \approx S^2$ proof check

I wanted to give a whole proof of this fact as I was not able to find a detailed one myself. I have the feeling that such a proof has been asked quite frequently by several users and I hope this may ...
3
votes
2answers
43 views

Proof of $a^x ≥ x+1 \; \forall x \in \Bbb R \implies a=e$

I'm trying to prove the following : Let $a>0$ a real number. Then : $\quad a^x ≥ x+1 \;\; \forall x \in \Bbb R \iff a=e$ I managed to prove the '$\Longleftarrow$' part : $x≥0$ then ...
1
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1answer
26 views

Verification of proof of propositional logic

I made a proof for the following theorem. But I'm not completely certain that it's fully correct. Suppose $\phi$ is a propositional formula and that the two evaluations $v$ and $w$ are equal for ...
1
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1answer
40 views

Proving $F(f)=\{x\in \mathbb R| f(x)=1\}$ isn't an injection

Let $$F: \mathbb N ^{\mathbb R}\to \mathcal P(\mathbb R)\\ F(f)=\{x\in \mathbb R| f(x)=1\}$$ Prove that $F$ isn't an injection. My attempt: Suppose $F(f_1)=F(f_2)$ and show that $f_1\neq ...
2
votes
0answers
61 views

Proving the Urysohn's metrization theorem by using the Nagata-Smirnov's metrization theorem

I need to prove the Urysohn's metrization theorem by using the Nagata-Smirnov's metrization theorem. Urysohn's metrization theorem: Every regular second-countable topological space is ...
2
votes
2answers
550 views

Least Upper Bound of 2 Sets

Let $A+B = \{a+b | a \in A, b \in B\}$ and denote the least upper bound of a set $X$ as $lub(X)$. Show that $lub(A+B) = lub(A) + lub(B)$. I'm finding this problem a little confusing. I've ...