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

learn more… | top users | synonyms

0
votes
2answers
52 views

Inequality with limsup from baby Rudin

For any two real sequences $\{a_n\}, \{b_n\},$ prove that $$\limsup_{n\to \infty}(a_n+b_n)\leqslant\limsup_{n\to \infty}a_n+\limsup_{n\to \infty}b_n$$provided the sum on the right is not of the form ...
1
vote
0answers
48 views

Why the proof isn't complete?

I'm going through some complex analysis exercises and found one with which I have some problems: For all real $y$, $$\int\limits_{-\infty}^\infty e^{-(x+iy)^2}dx = \int\limits_{-\infty}^\infty ...
3
votes
2answers
63 views

If a chain complex is homotopy equivalent to its homology, is it split?

Setup and conventions: Let $C_*$ be a chain complex of $R$-modules over some ring $R$, with boundary map $d$. The chain complex is said to be split if there exist $R$-linear maps $s: C_*\rightarrow ...
1
vote
2answers
38 views

Is the replacement theorem true for conditionals?

I read about the replacement theorem in Kleene's intro to logic which is as follows: If $\vDash(A\sim B)$ then $\vDash(C_A\sim C_B)$ where $C_A$ is a formula containing formula $A$ and $C_B$ is ...
2
votes
1answer
64 views

Can I assume that the dimension of a vector space is always non-negative?

I'm trying to prove that if $V$ is finite-dimensional and $U_1,...,U_m$ are subspaces of $V$, then $\dim(U_1+...+U_m)\le \dim U_1+...+\dim U_m$ through induction. For $m=1$, the inequality is trivial ...
0
votes
1answer
50 views

Verification of a proof involving metric spaces.

I've got this problem: If $(X,d)$ and $(X,d')$ are homeomorphic metric spaces, then they have the same convergence sequences. However, there exists homeomorphic metric spaces $(X,d)$, $(X,d')$ ...
0
votes
1answer
27 views

Doubt about the proof on uniqueness of saturated model

A standard proof for the fact that any 2 saturated models of the same cardinality are isomorphic can be found here But I have doubt about this. Specifically, the construction of the next partial ...
1
vote
0answers
25 views

Lie group is parallelizable

While going through the proof given by Alex Youcis at http://math.stackexchange.com/a/308798/86801 , I found the part where the local representation of the map $\ \Phi\ $ is shown to be the identity ...
1
vote
2answers
56 views

Critique my proof from munkres product topologies?

I've been going through Munkres' book on topology on my own, and I just struggled through the proof of 10d) from chapter 2 section 19. I've never had a chance to show one of my proofs to anyone, so I ...
0
votes
1answer
36 views

Show that if $\{X_n\}$ is a Markov Chain

Show that, if $\{X_n\}$ is a Markov Chain then $$P(X_n=j\mid X_k=l,X_m=i)=P(X_n=j\mid X_m=i),0\leq k<m<n$$ What I did is $$P(X_n=j\mid ...
0
votes
2answers
23 views

If $G$ is a simple, no loops graph, with n vertices and e edges, whose vertices have degree k or k+1 then G has $n_k$ vertices.

Question: Decide if the following expression is true or false. Prove or give a counterexample. If $G$ is a simple, no loops graph, with $n$ vertices and $e$ edges, whose vertices have degree ...
1
vote
1answer
53 views

A Proof in Elementary Set Theory

I apologize in advance for any lack of clarity in my mathematical symbols; I'm beginning to learn how to use this site. I'm working through "Introduction to Analysis" by Rosenlicht and he presents an ...
1
vote
1answer
51 views

Whether a real number is a dyadic rational iff its binary expansion terminates?

In self-studying a textbook on computability theory, I found that many of the exercises depend on the following factlet: A dyadic rational is a rational number whose denominator is a power of two, ...
1
vote
1answer
46 views

Find the sign of $a,b,c$ in $ax^2+bx+c$ given the graph and a coordinate on it.

So my first approach was that, we see that there are $2$ roots. And one is negative and one is positive. $a$ would be evidently positive. The positive one's modulus is bigger than the negative ...
0
votes
0answers
23 views

Simple question about asymptotic notation

I'm quiet new to the asymptotic world, so apology in advance if this question seems too trivial for you experts. Given $\frac{2kn2^{-k}}{E[X]}.$ As $k \sim 2\text{log}_{2} n$, the numerator is ...
1
vote
1answer
21 views

How is this an application of the independence property of events?

I'm currently working my way through Klenke's book on probability theory and do not understand a step in his proof of the Borel-Cantelli lemma (Theorem 2.7): The assertion is that for an independent ...
1
vote
0answers
35 views

Uniform Continuity of $\frac {1}{x}$ on [$a, \infty$) for positive $a$

$\frac {1}{x}$ behaves nicely in that it's monotone and the derivative is monotone also. So on [$a,\infty$] it can be seen that the $\delta$ which will work everywhere is the $\delta_1$ at the end ...
0
votes
1answer
31 views

The weak star convergence of Jordan decomposition

Given $\mu_n$ and $\mu$ finite signed Radon measures on the domain $\Omega$. We assume $\mu_n\to \mu$ in weak* sense, i.e. $\int_{\Omega}\phi \,d\mu_n \to \int_{\Omega} \phi\, d\mu$ for all test ...
1
vote
1answer
84 views

A proof of the existence of real root for odd degree real polynomials by induction

After thinking over the flaw in my argument of regarding the proof that every odd degree polynomial with real coefficients has at least one real root, I started devising another proof. The brief idea ...
1
vote
1answer
35 views

Use the Well Ordering Principle to prove that every finite, nonempty set of real numbers has a minimum element

This is a textbook problem. Here's my "proof": Assume for contradiction there exists a finite, nonempty set of real numbers which doesn't have a least element, call it $C$; suppose there are $n$ ...
0
votes
1answer
29 views

Proof given A,B is invertible, involving transposes

(Long time observer, first time asking a question, so excuse me if I get any of the rules wrong) I am having trouble wrapping my head around this problem and presenting the proof. If I know A, B is ...
0
votes
1answer
19 views

Proof that the fractional knapsack problem exhibits the greedy-choice property

I have the following problem: Prove that the fractional knapsack problem has the greedy-choice property. The greedy choice property should be the following: An optimal solution to a problem ...
1
vote
1answer
22 views

Determining Injectivity, surjectivity, bijectivity, and inverses

I was given a question that begins like this. Suppose that $A$ is the set $\{a,b,c\}$ (these are just names for some three elements - you don't know anything about $a,b,$ or $c$). Consider the ...
0
votes
0answers
29 views

How to prove $x^m = O(e^x)$ for any $m \gt 0$?

My attempt: It's true for $m = 1$ clearly. Now assume true for $m=1\dots M-1$. Then $x = O(e^x)$ and $x^{M-1} = O(e^{M-1})$. Lemma: if $f = O(g)$ and $f' = O(g')$ then $ff' = O(gg')$. Proof: $f = ...
2
votes
0answers
37 views

Help with Definition of Limits (Finding a delta given epsilon)

The problem says: Find a $\delta$ such that $|f(x)-l| < \epsilon$ for all x satisfying $0 < |x-a| < \delta$ when $f(x) = x^4; l = a^4$. What I did so far was $|x^4-a^4| < \epsilon$ so ...
0
votes
1answer
14 views

Question regarding the proof of 'If $W_1\leqslant W_2$ then $W_2^\perp \leqslant W_1^\perp$'

Let $W_1, W_2$ be subspaces in a finite-dimensional inner space $V$. Show: If $W_1\leqslant W_2$ then $W_2^\perp \leqslant W_1^\perp$. My proof Let $a,b \in W_2^\perp$ then $(\forall w\in ...
4
votes
1answer
76 views

How to show that $|\exp(z)-1|\le2|z|$ for $|z|\le 1$

How to show that $|\exp(z)-1|\le2|z|$ for $|z|\le 1$ ...
2
votes
2answers
61 views

On the inner workings of induction?

I always had some doubts on the inner workings of induction. So I decided to make a little experiment. I am familiar with the proof that the sum of the first $n$ integers is $\cfrac{n(n+1)}{2}$ so I ...
1
vote
1answer
142 views

Can a method related to “Weisfeiler-Lehman Method” provide better time complexity for Graph Isomorphism than existing result?

Cai-Furer-Immerman showed that the W-L(Weisfeiler-Lehman ) hierarchy cannot distinguish general graphs except at linear dimension. Even besides CFI's result, there is good reason to believe that ...
3
votes
2answers
91 views

Is my proof for the existence of roots of an odd-degree polynomial correct?

$\color{crimson}{\text{Problem}}$ Show that if $f:\mathbb{R}\to\mathbb{R}$ be a polynomial of odd degree with real coefficients then it has at least one real root. $\color{green}{\text{Proof}}$ Let ...
1
vote
2answers
50 views

$\liminf_{n\to\infty} \frac{\varphi(n)}{n} = 1$, not $0$

Let $\varphi(n) = \sharp\{1\leq x \leq n : (x,n) = 1\}$. Then $\liminf_{n\to\infty} \frac{\varphi(n)}{n} = 1$. My attempt: $\inf_{k\geq n}\frac{\varphi(k)}{k} \leq 1$ since for $n = 2$, this ...
2
votes
6answers
161 views

How do I prove this $\frac{dx^n}{dx}=nx^{n-1}$ is true for every $n\geq 1$ to convince my students?

let $p_n(x)=x^n$ be a polynomial of degree $n$. I need help to be able to explain to my students why the derivative of $p$ is defined as follows: $$ p_n'(x)=\frac{dx^n}{dx}=nx^{n-1} $$ for every ...
0
votes
1answer
16 views

Analysis Proof- different conditions.

A continuous function on $[a,b]$ is also uniformly continuous on $[a,b]$. The following tries to illustrate what happens when the interval is not closed: Show: $f(x) = \frac{1}{x} $ is not ...
0
votes
1answer
41 views

Is this alternative proof of Theorem 3.7 (“Baby” Rudin, Ch. 3) correct and, if so, well written?

Rudin, in his Principles of Mathematical Analysis, proves the following theorem: The subsequential limits of a sequence $\{p_n\}$ in a metric space $X$ form a closed subset of $X$. I've tried to ...
0
votes
0answers
27 views

Limit Points and Convergence of Sequences.

Let $E \subseteq \mathbb{R}$ (or $\mathbb{C}$). A point $p \in \mathbb{R}$ (or $\mathbb{R}$ ) is called a limit point of $E$, if $\forall \epsilon > 0$, $\exists z \in E$ such that $0 < |z − p| ...
7
votes
1answer
58 views

$a \in G$ commutes with all its conjugates iff $a$ belongs to an abelian normal subgroup of $G$ [duplicate]

Let $a\in G$, where $G$ is a group. Prove that $a$ commutes with each of its conjugates in $G$ if and only if $a$ belongs to an abelian normal subgroup of $G$. This is what I did: $"⟹"$ First, ...
0
votes
1answer
34 views

What is wrong with this inductive proof?

I have found a startling proof by induction which is clearly wrong. Let L(n) represent Lucas numbers. L(0)=2, L(1)=1 L(n) = L(n-1) + L(n-2) Let F(n) denote a Fibonacci number. F(0) = 0, F(1) = 1, ...
2
votes
1answer
66 views

The proof of theorem 3.19 from baby Rudin

If $s_n\leqslant t_n$ for $n\geqslant N$, where $N$ is fixed, then $$\liminf_{n\to \infty} s_n\leqslant \liminf_{n\to \infty} t_n$$ $$\limsup_{n\to \infty} s_n\leqslant \limsup_{n\to \infty} t_n$$ ...
1
vote
5answers
49 views

Help with proof that $\sum_{n \in \Bbb{N}} \frac{1}{an + b}$ also diverges?

We know that $\sum_{n \in \Bbb{N}} \frac{1}{n}$ diverges. So it seems likely that $\sum_{n \in \Bbb{N}} \frac{1}{a n + b}$ will for any real $a, b$. I'm having trouble proving it just for the ...
2
votes
2answers
31 views

Analysis Limit- Function Proof

A) For every sequence ($p_n$) in $E$ such that $p_n \not= p$ and $p_n \rightarrow p$ as $n \rightarrow \infty$ we have that $f(p_n) \rightarrow l$ as $n \rightarrow \infty$ . ($E \subseteq ...
0
votes
1answer
28 views

Help with Spivak's Calculus Chapter 3 Problem 6

It says: Show that the straight line through $(a,b)$ with slope $m$ is the graph of the function $f(x) = m(x-a) + b$. Since the slope in a graph of a line is determined by using similar triangles, ...
0
votes
0answers
22 views

Complex Sequence Convergence

Claim : If complex sequence $z_n$ converges then $|z_n|$ converges Proof: Let $z_n =x_n +i y_n$ where $x_n$ and $ y_n$ are real sequences. If $z_n $ converges to $(L_1+i L_2)$ $ \forall \epsilon ...
3
votes
2answers
45 views

A plane contains a set of marked points, such that any three can be covered by a unit disk. Prove that the entire set can be covered by a unit disk.

A set of points is marked on the plane, with the property that any three marked points can be covered with a disk of radius 1. Prove that the set of all marked points can be covered with a disk ...
0
votes
1answer
26 views

Show that $\left \{ \bigcup_{i\in I}A_{i}:I\subseteq \{1,\dots, n\} \right \}$ is a $\sigma$-algebra

Let $\{A_{i}\}_{i = 1}^{n}$ be a family of pairwise disjoint subsets of $X$. It is said that $$\mathcal{F}:=\left \{ \bigcup_{i\in I}A_{i}:I\subseteq \{1,\dots, n\} \right \}$$ is a $\sigma$-algebra. ...
4
votes
1answer
50 views

Do proof assistants like Coq really need to actually perform computations to prove n <= m, or is there a more optimal algorithm?

For example, trying to prove that 100,000 <= 1,000,000. But Coq has a stack overflow, meaning it's actually trying to perform the 100k computations. ...
1
vote
1answer
27 views

Order of a corrector-predictor method

Given an explicit method: $$ x_{i+1} = x_i+ h \Phi(t_i,x_i,h) $$ as predictor method and an implicit method: $$ x_{i+1} = x_i + h \Psi(t_i,x_i,x_{i+1},h) $$ as corrector method, it follows that $$ ...
1
vote
2answers
41 views

Calculus Spivak. Chapter 1. Question 1. (i) or are there many ways of skinning a cat

I'm taking on Spivak's Calculus a little later on in life via self-study as i'm looking to improve my CS abilities and have always been interested in Maths but unfortunately didn't have the chance ...
1
vote
1answer
22 views

If $S\leq G$, prove that $S\unlhd G \iff \gamma (S) \leq S$ for every conjugation $\gamma$

If $S\leq G$, prove that $S\unlhd G \iff \gamma (S) \leq S$ for every conjugation $\gamma$ I have proven the forward direction but I am not sure that the way I prove the converse is true. Let $g\in ...
1
vote
4answers
62 views

Prove that if $A$, $B$ are countable, then $A \times B$ is countable?

Is $A\times B$ referring to the axis here? So an $X$ and $Y$ coordinate plane? $A$ is countable, therefore a bijection occurs from $A \rightarrow \mathbb{N}$. $B$ is countable, therefore a bijection ...
0
votes
2answers
29 views

Prove or disprove the following asymptotic relations

$P(x) = 2^x$ Prove or disprove that $p(n^3 + 4) \in O\left(p\left(n^3\right)\right)$ $2^{(n^3 + 4)} \in O(2^{n^3})$ $\lim_{n \rightarrow \infty} \space \frac{2^{n^3 + 4}}{2^{n^3}}$ using ...