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

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2answers
54 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 … ...
0
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0answers
25 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 ...
1
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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 $\widetilde{H}_0(X;\mathbb{Z})\cong \mathbb{Z}$ ...
0
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0answers
21 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 ...
0
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0answers
49 views

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

I want to show the following statement: If $u \in W^{1,p}(I) \cap C_c(I)$ 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 are ...
1
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2answers
42 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
24 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
30 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
61 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
26 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?
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2answers
35 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: ...
0
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0answers
26 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 $E$ ...
0
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2answers
19 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 ...
3
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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 ...
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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 ...
1
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2answers
36 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
58 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 ...
0
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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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1answer
21 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
votes
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 ...
1
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0answers
38 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
55 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. ...
23
votes
2answers
527 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
49 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 ...
3
votes
1answer
25 views

Norm of the operator on Hilbert Space $l_2$

Suppose $l_2 = \{x= (x_n) | \sum x_n^2 < \infty \}$ is a Hilbert Space and $T( (x_n))= (x_2 -x_1, x_3 - x_2, \dots , x_n-x_{n-1}, \cdots )$. Which of the followings are true a) $||T|| =1$ b) ...
0
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1answer
44 views

non-zero divisors in a ring

I am asked to show the following: $ab$ is a non-zero divisor of $R$ if and only if $a$ and $b$ are both non-zero divisors of $R$. $\Rightarrow)$ Suppose $ab$ is a non-zero divisor of R. Then ...
2
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0answers
29 views

Functions disagreeing on a set of measure zero - Proof verification

I was asked to prove that if $f\in\mathscr{R}$ on some compact rectangle $Q\subset\mathbf{R}^{n}$ and if the set $D=\lbrace \mathbf{x}:\mathbf{x}\in Q,\,f(\mathbf{x})\neq 0\rbrace$ has measure zero, ...
2
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1answer
33 views

If there are nonzero elements $a$ and $b$ in $A$ such that $(a+b)^2 = a^2 + b^2$, then $A$ has characteristic 2.

Let $A$ be a finite integral domain. If there are nonzero elements $a$ and $b$ in $A$ such that $(a+b)^2 = a^2 + b^2$, then $A$ has characteristic 2. I was thinking, if $(a+b)^2 = a^2 + 2ab + b^2 = ...
1
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1answer
22 views

Determining whether or not these spaces are convex

Consider $\{(x,y) \in \mathbb{R}^2: |x| +y^2\leq 5\}$ and $\{ (x,y) \in \mathbb{R}^2: y\geq x^2,y\leq e^{-x^2}\}$. Determine whether or not these two are convex sets. I have used the visual ...
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1answer
33 views

Proof of probabilty notation

G is the event of a customer buying a goldfish from a pet shop and T is the event of a customer buying a tortoise at a pet shop P (T) is twice P (G), P (G nor T) =0.42 and G and T are independent i) ...
1
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0answers
31 views

Use the arithmetic-geometric inequality for this list to deduce the arithmetic-geometric inequality for $n$.

Suppose that $n$ is not a power of two. Let $2^k$ be a power of $2$ that exceeds $n$ and consider the list $$a_1,\dots,a_n,\underbrace{A,A,\dots,A}_\text{$2^k-n$ times}$$ of length $2^k$. Use the ...
0
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0answers
48 views

Is there any difference between “for any” and “for all”?

When we prove something, we use mathematical symbol ∀ to stand for "for all." Does it make any difference if we use same symbol for "for any."?
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1answer
31 views

Proof concerning isosceles triangles

In the triangle $ABC$ it is $AC = BC$ and $\alpha = \beta$. The points $D$ and $E$ are on the line through $A$ and $B$. Show that the triangle $CDE$ is isosceles. Hey there! Is it sufficient ...
0
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1answer
27 views

Proof verification: If $a_n\to L$ and $r<L$, there exists $n_0$ such that for all $n\ge n_0$, $a_n> r$.

Exercise. Prove that if $a_n\to L$ and $r<L$, there exists $n_0$ such that for all $n\ge n_0$, $a_n> r$. Attempt. Suppose not. Then for all $n_0\in \bf N$, we have that there exists ...
1
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0answers
25 views

How to make inductive step for a Fibonacci proof [duplicate]

I have to prove $F^2_{n−1} = F^2_n + F^2_{n−1}$ for any $n >=1$ by induction (for the Fibonacci sequence). For the basis step, I have: $n = 1; $ $F_{(1)-1} = F^2_{(1)} + F^2_{(1)-1} ->$ $ ...
0
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1answer
46 views

Proof check/ suggestion: The suspension of $S^n$

In one of my excercise sheets there was a remark saying that $$SX \approx S^{n+1}$$ where $SX$ denotes the suspension of $X=S^n$. So I tried to prove this on my own and would like to discuss my ...
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2answers
50 views

What is the denial of a statement in logic math?

I'm trying to get the hang of denials in logic in math. I would like to use these two examples: "Some people are honest and some people are not honest. (All people)" "No one loves everybody. (All ...
0
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0answers
10 views

Sufficient conditions differentiability in quadratic mean

I'm trying to show lemma 7.6 in van der Vaart "Asymptotic Theory" on the sufficient conditions for differentiability in quadratic mean of a probability density function but I have some doubts when it ...
7
votes
2answers
105 views

Proof verification for $\mathcal{P}(A) \cap \mathcal{P}(B) = \mathcal{P}(A \cap B)$

I propose here my proof for: $$\mathcal{P}(A) \cap \mathcal{P}(B) = \mathcal{P}(A \cap B)$$ $\Longrightarrow$ $$x \in \mathcal{P}(A) \land x \in\mathcal{P}(B)$$ $$x \subseteq A \land x \subseteq B$$ ...
0
votes
1answer
33 views

Proof Verification - All nonnegative reals have unique nonnegative square roots

I am asking simply to verify a proof since I don't think this is standard. In particular, this is a theorem in Apostol's Calculus book but I wanted to give it a go myself! Theorem: $\forall c \in ...
0
votes
1answer
36 views

Zero divisors within the ring of dual numbers

Let $\mathbb{R}(\epsilon) = \{ a + b \epsilon : a,b \in \mathbb{R} \}, $ where $(a + b\epsilon) + (c + d\epsilon) = (a+c) + (b+d)\epsilon$ and $(a+ b\epsilon) \cdot (c + d\epsilon) = ac + (ad + ...
1
vote
0answers
27 views

Prove $\sqrt s$ exists by the Intermediate Value Thorem?

I'm pretty sure the process is right, but I'm not sure if I've validly presented the proof. Here is what I wrote: Suppose that $s > 0$ and consider the function $f(x) = x^2$ on the interval ...
1
vote
1answer
23 views

Surjectivity and the non-existence of maps.

This question comes from Jacobson's Basic Algebra. It asks: Show that $S \overset{\alpha}{\to} T$ is surjective iff there exist no maps $\beta_1,\beta_2$ of $T$ into a set $U$ such that $\beta_1 ...
0
votes
1answer
29 views

Increasing real valued function whose image set is connected

Let $S = [0,1) \cup [2,3]$ and $f\colon S \rightarrow \mathbb R$ be such that $f(S)$ is connected . Which of the following are true: a) $f$ is discontinuous exactly at one point. b) $f$ is ...