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

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1
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0answers
17 views

Proving the continuity of the function $f(x)=x^\alpha$ through power series

The textbook I'm studying on presents the following two theorems: $a^x=\sum_{n=0}^{\infty}\frac{(\ln{a})^nx^n}{n!}$ The function $f(z)=\sum_{n=0}^{\infty}a_nz^n$ converges for every $z\in\mathbb{C}$ ...
6
votes
1answer
21 views

Verification for a block-determinant evaluation, and some further thoughts

First, I want some verification for the validity of my approach for this det evaluation question: If $A,B\in M_n(K)$, $K$ is a number field (in the sense that $\Bbb Q$ is the smallest possible ...
2
votes
1answer
28 views

Problem 20 chapter 3 from baby Rudin

Suppose $\{p_n\}$ is a Cauchy sequence in a metric space $X$, and some subseqeunce $\{p_{n_i}\}$ converges to a point $p\in X$. Prove that the full sequence $\{p_n\}$ converges to $p$. Proof: ...
3
votes
1answer
43 views

Prove $x^n < n^n 2^x$

Given that $$x < 2^x$$ is always true, use it to prove that $$x^n < n^n2^x$$ Here are the steps that I've taken so far: Reduce $$x < 2^x$$ to $$\log(x) < x$$ Then $$x^n < n^n2^x$$ ...
1
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1answer
21 views

Show that this piece-wise function defines a differentiable solution

Show that $y(x) = \begin{cases}-x^4 & x < 0, \\ x^4 & x \geqq 0 \end{cases}$ defines a differentiable solution of $xy'=4y$ for all $x$, but is not of the form $y(x)=Cx^4$.
2
votes
2answers
39 views

$\{a_n\} \to a$ iff $\limsup_{n \to \infty} \{a_n\} = \liminf_{n \to \infty} \{a_n\}$

It is clear that if $$\limsup_{n \to \infty} \{a_n\} = \liminf_{\to \infty} \{a_n\},$$ then $\{a_n\} \to a$, since we can just squeeze the terms in the middle. I understand that to prove the ...
0
votes
0answers
11 views

The martingale $M_t,\mathcal{F}_t$ is a martingale with respect to the filtration $\mathcal{F}_{t +}$

Let $M_t$ be a right continuous martingale with respect to the filtration $\mathcal{F}_t$. Can we say that $M_t$ is a martingale with respect to the filtration $\mathcal{F}_{t+}$? Attempt: We know ...
1
vote
1answer
26 views

Algorithm and top-points.

Problem: For an array $A[1],\dots,A[n]$, with $n\geq 3$, it holds that $$A[i+1]>\frac{A[i]+A[i+2]}{2},\qquad i\in \{1,2,\dots, n-2\}$$ That is, it holds that $$A[2]>\frac{A[1]+A[3]}{2},\dots, ...
1
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1answer
19 views

Proof Verification - $\exists a \in S (a\ge S_a)$

I wanted to prove that $\exists a \in S (a\ge S_a)$ where $S$ is an finite set of real numbers with order $n$ and $S_a$ is the average of the set. This is my proof so far: Assume $a_i = a_k, i,k ...
1
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0answers
6 views

Is the function $Z(t,q)$ progressively measurable?

The excercise is taken from Stroock and Varadhan Multidimensional diffusion processes chap 1 page 44 is the following To the first part, I reasoned as follows: Consider the function $F(q_1,q_3) = ...
0
votes
0answers
41 views

Fermat's theorem on two squares - what do I missing?

In Wikipedia there's a list of quite non-trivial and beautiful proofs of Fermat's two squares theorem. Actually I'm a bit surprised because this fact belongs to a very small set of mathematical fact ...
-2
votes
1answer
39 views

The cardinality of the integers is divisible by all prime numbers?

In this question Parity of members in a group I defined even members of a group $G$ as all members $b \in G : b \neq a^ca^{c+1}$ where $a \in G$ and $c \in \mathbb{N}$ . This follows from the fact ...
1
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2answers
62 views

Proving that if $p_{n}$ converges then $|p_{n}| $ converges

EDIT1: Prove using the definition of a converging sequence in a metric space, that the convergence of the sequence $\left \{ p_{n} \right \}_{n=1}^{\infty}$ implies the convergence of the sequence ...
2
votes
1answer
108 views

Elementary Twin Prime Attempt. [on hold]

There's a theorem somewhere that for sufficiently large $k$ there exists an infinite number of prime pairs with difference $2k$. Let $\ell$-prime pair mean a pair of primes separated by a distance of ...
0
votes
1answer
22 views

Approximation of continuous functions by Bernstein polynomials

Recently a professor show me the following heuristic to provide approximations of continuous functions by polynomials: Let $P_n(x) = \sum_{k=0}^{n} {n \choose k} f(\frac{k}{n}) x^k (1-x)^{n-k}$. ...
1
vote
1answer
38 views

Field homomorphism induces an isomorphism between their prime subfields

So the question is: Let $\sigma$: $F_1 \xrightarrow[]{} F_2$ be a homomorphism where $F_1$ and $F_2$ are fields. Show $\sigma$ induces an isomorphism between their prime subfields and, in ...
1
vote
2answers
69 views

What's the importance of proving that $0,1$ are unique?

I had a course in the construction of numbers last semester. I understand the potencial of most of the proofs, for example: I guess I can answer decently why commutativity is important. But when it ...
1
vote
1answer
33 views

Is my work correct? (Easy problem, confidence intervals)

The r.v. $X$ represents the time taken by a computer in company $1$ in order to perform a certain job, and $Y$ represents the same thing but for company $2$. A sample of $n_X = 12$ computers are taken ...
3
votes
1answer
22 views

All closed rational rays measurable implies $f$ measurable

Is the following proof correct? Let $f: X \to \mathbb{R}$ where $X$ is a measurable space. Suppose $\{x: f(x) \geq r\}$ is measurable for each $r \in \mathbb{Q}$. Then, $f$ is measurable. Proof: ...
1
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1answer
25 views

Show $R \setminus S$ is a union of prime ideals

I'm stuck on the following question: Let $R$ be a commutative ring with $1$, and $S \subseteq R$ a saturated multiplicative set (that is, $1 \in S$ and $x, y \in S$ if and only if $xy \in S$). ...
0
votes
1answer
18 views

If $x,y,a_1,\ldots,a_n\in\mathbb Z$ and $a_1,\ldots,a_n\in[x,y],$ then $a_1=x,a_2=x+1,\ldots,a_n=y$ (proof verification)

I recently solved a problem in which I used the following fact. If there are exactly $n$ integers in the interval $[x,y]$ ($x,y\in\mathbb Z$) and $a_1,a_2,\ldots,a_n$ are integers of that interval ...
1
vote
1answer
34 views

Alternative proof of the Riemann Sum Theorem using Mean Value Theorem for Integrals.

I've been reviewing proofs for a couple of calculus theorems and as I was trying to recall the proof of the Riemann Sum Theorem which uses Lower Sums and Upper Sums I came up with an idea to prove it ...
0
votes
2answers
37 views

Let $X$ and $Y$ be finite sets. Then $X \cup Y$ is finite and $| X \cup Y| \leq |X| + |Y|$.

Let $X$ and $Y$ be finite sets. Let us assume that they are distinct at least, for otherwise $X \cup Y = X$ and $X$ is finite. Also let us assume that $X$ has cardinality $n$ and $Y$ has cardinality ...
2
votes
4answers
42 views

Prove $\log(x) < n(x)^{1/n}$, for all positive integer values of $n$, and $x > 0$

Given that $$lg(u) < u$$ is always true, how do we use that to prove that $$lg(x) < n(x)^\frac 1n$$ These are the steps that I have taken so far: $$1: lg(x) < n(x)^\frac 1n$$ $$2: \frac ...
1
vote
0answers
45 views

On n! divided by a product of primes and related questions

We have the following Definition 1. For integers $n\geq 1$ we define $$f(n) = \begin{cases} 1, & \text{if $n=1$} \\[2ex] \frac{n!}{\prod_{p\leq n}p}, & \text{if $n>1$} ...
5
votes
1answer
58 views

Proof of Vandermonde's Identity using a “different approach” using complex integration

Hi I'd like to know if the following proof of Vandermonde's Identity is correct (is really easy): Let $m,n,r$ be natural numbers such that $r\le \min \{m,n\}$. The Vandermonde's Identity gives us ...
-1
votes
1answer
41 views

I want to appeal this problem from an exam in Linear Algebra I, do you think its appealable? [on hold]

I have the follow question : Let $U_1, U_2, W$ are linear spans of linear space $V$ while V is finite. Proof: If $$U_2 \cap W \neq \{0\}$$ $$U_1\cap W\neq \{0\}$$ $$U_1 \cap U_2=\{0\}$$ Then $dimW ...
2
votes
1answer
26 views

Understanding pasting lemma proof

Let $A$ and $B$ be both open or closed subsets of a topological space $X$ such that $A \cup B = X$. Let $f: A \to Y$ and $g: B \to Y$ be continuous such that $f = g$ for all $x \in A \cap B$. Prove ...
1
vote
2answers
26 views

Prove: For every $\epsilon > 0$, there exists a $\delta >0$ such that $1 - \delta < x < 1 + \delta$ implies that $2 - \epsilon < 7 -5x< 2 + \epsilon$

So far this is what I have Let $$ \delta = \frac{\epsilon}{5} $$ So, if we start with $1 - \delta < x < 1 + \delta$ \begin{align} &\Rightarrow -5 + 5\delta < -5x ...
7
votes
1answer
70 views

Functional equation: Show $0\le f(n+1)-f(n)\le 1$ and find all $n$ such that $f(n)=1025$.

The function $f:\mathbb{N}\to \mathbb{R}$ satisfies all of $$\begin{align}f(1)&=1, \\ f(2)&=2,\\ f(n + 2) &= f(n + 2 − f(n + 1)) + f(n + 1 − f(n)) \tag{1} \end{align}$$ Show ...
2
votes
0answers
35 views

For any Subspace $A$ of a Path-Connected Space $X$, we have $H_0(X, A)=0$.

I recently learnt about relative homologies and am wondering if the following is true: Statement: Let $X$ be a path-connected topological space and $A$ be a non-empty subspace of $X$. Then $H_0(X, ...
3
votes
0answers
22 views

Help in step of the proof of Burnside's $p^aq^b$ theorem (Doerk-Hawkes book)

I'm reading the proof of the $p^aq^b$ Burnside's theorem from the book Finite soluble groups by Doerk and Hawkes. The fifth step of the proof says 2.5. Let $M$ and $H$ be maximal subgroups of $G$ ...
5
votes
0answers
38 views

For finitely generated free abelian groups $A,B$ if there is an onto homomorphism $A \to B$, then $\operatorname{rank}(A) \geq \operatorname{rank}(B)$

$\newcommand{\rank}{\operatorname{rank}}$For two finitely generated, free abelian groups $A,B$ prove that if there is an onto homomorphism $A \rightarrow B$, then $\rank(A) \geq \rank(B)$ Assume that ...
1
vote
1answer
18 views

Cyclic Groups - $a^k = e \text{ iff } n|k$

I saw this proof in the book on Abstract Algebra. Here is part of it: Let $G$ be a cyclic group of order $n$ and $a$ is the generator of $G$. Then $a^k = e \iff n|k$ Proof: Suppose $a^k=e$. By the ...
1
vote
1answer
17 views

Proof check on $I (\lim \sup E_n(w)) = \lim \sup I_{E_n}(w)$, where $I$ is the indicator function.

I proceed by cases: $ \lim \sup I_{E_n}(w) := \lim_m \downarrow \{ \sup_{n>m} I_{E_n}(w) \}$ this is $0$ only if $w \not \in E_n \forall n > m$. At the right of the equality we have ...
1
vote
1answer
30 views

Validity of my proof by contradiction of converse of Pythagorean Theorem.

So in, $\triangle ABC$ it is given that $AB^2=AC^2+BC^2$. Let us assume that $\angle C\neq{90}^{\circ}$. And let us make a perpendicular $AD$ to $BC$. Now, by the Pythagorean Theorem, in $\triangle ...
2
votes
1answer
31 views

Showing that $\mathbb{R}$ is a metric space with the metric $d(x,y)=|x-y|$

I want to show that $\mathbb{R}$ is a metric space with the metric $d(x,y)=|x-y|$. So three properties of the metric space $d(x,y)$ in general needs to be satisfied. My work: Let $x,y \in ...
3
votes
0answers
27 views

My proof that the Alexandroff double circle not second-countable

I'm hoping someone can comment on if my logic on the Alexandroff double cirlce not being second countable is right. The Alexandroff double circle has underlying set $C = C_1 \cup C_2$ where $C_i = \{ ...
1
vote
1answer
21 views

Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one.

Let $X,Y$ be nonempty sets and $f:X\to Y$ be a function. Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one. My approach is by ...
0
votes
0answers
27 views

Topology generated by basis equals intersection of all topologies that contain A

Here is my proof I was wondering any critiques to my proof. If A is a basis;The topology generated by A equals collections of all unions of elements of A that is $\tau = \bigcup_{i \in I: B_i \in ...
4
votes
0answers
57 views

A quick Galois Theory question

Let $\mathbb{F}_q$ be a finite field of order $q$ where $q$ is a prime power. For any $d \in \mathbb{N},$ we have an inclusion $\mathbb{F}_{q^d} \subseteq \overline{\mathbb{F}}_q.$ Both ...
1
vote
2answers
15 views

Partitioning a number as a sum of $k$ non-zero numbers, but order does not matter

I would like some confirmation regarding my logic here, which I feel is 'suspiciously straightforward'. Say I wish to express a number as the sum of $10$ non-zero numbers, where order does not ...
0
votes
1answer
16 views

Normal Matrix with Real Eigenvalues is Hermitian

Let $A$ be a normal matrix. Then I want to show that, if $A$ has real eigenvalues, $A$ is Hermitian. (Notation: * denotes the complex conjugate, T denotes the transpose, and $\dagger$ denotes the ...
2
votes
3answers
94 views

Proving $\sin^2(x) + \cos^2(x) =1$ using calculus

Ok so the book in which I found this doesn't say mention the trigonometric functions by name but the question is: Let $s(x)$ and $c(x)$ be functions satisfying $s'(x)=c(x)$ and $c'(x)= -s(x)$ for ...
0
votes
1answer
16 views

Is non-paralellism transitive?

I've been trying to check if non-paralellism is transitive. At the moment, I know it's symmetric. But I have no idea on how to prove that it's transitive. I did the following: $$(a \not\parallel b) ...
1
vote
1answer
38 views

Exercise from (Baby) Rudin (Chapter 3, exercise 13): is this proof correct? Is it well-written?

The problem is the following: Prove that the Cauchy product of two absolutely convergent series converges absolutely. Here is my attempt: Let $s_n=\sum^n_{k=0}a_k$ and $t_n=\sum^n_{k=0}b_k$ be ...
2
votes
1answer
36 views

Proving that limit of a sequence is 0 from definitions.

I had this question in a test: Use the definition of limit in order to prove that if $\{a_n\}$ (n goes from 1 to infinity) is a sequence of real numbers such that $\lim_{n\rightarrow \infty} a_n^2 ...
1
vote
0answers
22 views

The restriction of a discontinuous linear functional to any open set is surjective.

Problem. Let $X$ be a topological vector space and $f:X\to\mathbb{K}$ a linear mapping. Prove that if $f$ is discontinuous, then $f(A)=\mathbb{K}$ for all nonempty open set $A\subset X$. I'd like ...
2
votes
1answer
17 views

If $N$ is nilpotent of index $n\geq 2$ but $N^{n-1}\neq 0$ then there's no $A$ such that $A^2=N$

Let $N\in M_{n\times n}^{\mathbb{C}}$ a nilpotent matrix of index $n\geq 2$. Prove: if $N^{n-1}\neq 0$ then there does not exist a matrix $A\in M_{n\times n}^{\mathbb{C}}$ such that $A^2=N$. My ...
1
vote
1answer
73 views

Is this a valid proof of this math challenge problem?

From a fixed point P not in a given plane, three mutually perpendicular line segments are drawn terminating in the plane. Let a, b, c denote the lengths of the three segments. Show that ...