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

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3
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0answers
40 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
14 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
83 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
14 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
34 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
31 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 ...
0
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0answers
19 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
71 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 ...
0
votes
1answer
19 views

Find a criterion for divisibility

Find a criterion such that $\displaystyle\sum_{i=1}^ni$ divides $\displaystyle\prod_{i=1}^ni^2$ for $n\in\mathbb N$. What I have done so far, $\displaystyle\sum_{i=1}^ni=\frac{n(n+1)}{2}$ and ...
2
votes
2answers
144 views

What makes it legitimate to multiply both sides?

Having the proof of the cancelation law for multiplication: $$cb=ab$$ $$(cb)b^{-1}=(ab)b^{-1}\tag{Inverse}$$ $$cbb^{-1}=abb^{-1}\tag{Associativity}$$ $$c\cdot 1=a\cdot 1\tag{Indentity}$$ $$c=a$$ ...
3
votes
5answers
102 views

Proof check: $(4n)!$ is divisible by $2^{3n}3^{n}$

Question: Show that $(4n)!$ is a multiple of $2^{3n}3^{n}$ for all $n$. Proof: It's easy (involves kinda messy calculation tho) to show by induction that $(4n)!$ is a multiple of $2^{3n}$. Now, since ...
0
votes
2answers
34 views

Weak convergency vs strong convergency in Hilbert space

Let $\mathcal{H}$ be an Hilbert space and let $(x_n)_n \subset \mathcal{H}$ be a sequence s.t. $$ x_n \rightharpoonup x ~~~,~~~ \| x_n \| \to \|x\| $$ We want to show that $ x_n \to x $. Now, I ...
0
votes
1answer
31 views

Lower bound for $\Pi(n)$ - viability of probabilistic theory

Can somebody check the validity of my arguments below, and tell me why its wrong or right? Consider the sequence of non-negative integers. Let $a_0=0, a_1=1, ..., a_i=i,...$ Divisiblilty of $a_i$ ...
3
votes
0answers
17 views

Iterated circumcenters - proving collinearity and establishing distance ratios

Let $P_0, P_1, P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle ...
1
vote
1answer
33 views

Find all primes $p$ for which $x^2+2x+4\equiv 0 \pmod p$ is solvable. Am I correct?

Getting ready for an exam, I would like to focus on the correctness of my solution, final results and assumptions, and would appreciate any comment regarding it or even additional ...
1
vote
2answers
28 views

Proving that $a^{b}$ is rational (Elementary number theorey) [duplicate]

Prove that there exist irrational numbers $a$ and $b$ such that $a^{b}$ is rational. What i tried Prove by contradiction I assume the statement For all rational numbers $a$ and $b$ such that ...
1
vote
0answers
42 views

Problem 14 from Baby Rudin chapter 3

Let $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\dfrac{s_0+s_1+\dots+s_n}{n+1},\quad a_n=s_n-s_{n-1} \quad\text{for} \quad n\geqslant 1$$ Assume ...
1
vote
1answer
36 views

$\gamma=(\psi \implies \phi)$ is a tautology $\equiv \psi$ is a contradiction or $\phi$ is a tautology.

Prove that: If $\psi,\phi$ are formulas such that $\text{VAR$(\psi)$} \cap\text{VAR$(\phi)$}=\emptyset$. Then $\gamma=(\psi \implies \phi)$ is a tautology $\equiv \psi$ is a ...
0
votes
1answer
12 views

Show that a line is tangent to a circle in the extended complex plane.

The straight line $l$ in the extended-complex plane pasess through $2+i,2+2i$.The circle $C$ centered at $-1-2i$ with radius $3$. First, I find the parametrization of the straight line which is $$z = ...
3
votes
4answers
78 views

Proof of $(A\cup B)-(A\cap B)=(A-B)\cup(B-A)$

I was trying to prove $(A\cup B)-(A\cap B)=(A-B)\cup(B-A)$ and came across issues in translating (pertaining to what I did with $\emptyset$) and got through the proof but was doubting its accuracy so ...
7
votes
3answers
139 views

Prove that $\det(I-CD)=\det(I-DC) $

Let $C$ and $D$ be matrices such that $DC$ and $CD$ are square matrices of the same dimension. How can one prove that $\det(I-CD)=\det(I-DC)$? This is my approach to the question. I am not sure ...
0
votes
0answers
18 views

Generalized Associative Property (Proof Verification)

I am really confused about Associative property and Generalized associative property. I am not sure of my proof, and I have a feeling that it is not correct. Would be happy if someone can tell me what ...
3
votes
2answers
31 views

Proving a function $F$ is surjective if and only if $f$ is injective

Problem: Let $X$ and $Y$ be non-empty sets and let $f: X \rightarrow Y$ be a function. Then we can define $F: P(Y) \rightarrow P(X)$ by \begin{align*} F(B) = f^{-1}(B) \qquad \text{for all} \ B \in ...
5
votes
0answers
43 views

A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have?

A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have? options given: (a) either $1$ or $2$ (b)$1$ or $3$ (c)either $2$ or ...
0
votes
1answer
36 views

Check whether this is indeed a counterexample

Let $A,B \subset \mathbb{R}$; let $Q := A \times B$; and let $f: Q \to \mathbb{R}$ be bounded. The problem is to give a counterexample to the proposition that if the Riemann integral $\int_{Q}f$ ...
2
votes
2answers
85 views

If $\gcd(ab,c)=d$ and $c|ab$ then $c=d$

For all positive integers $a$, $b$, $c$ and $d$, if $\gcd(ab, c) = d$ and $c | ab$, then $c = d$. Need help proving this question, I know that $abx + cy = d$ for integers $x,y$ and that $c|ab$ can be ...
2
votes
0answers
24 views

Characterization of graphs of maps between smooth manifolds

Theorem 6.52 in Lee's Introduction to Smooth Manifolds, 2nd ed., says Suppose $M$ and $N$ are smoothe manifolds and $S \subset M \times N$ is an immersed submanifold. Let $\pi_M$ and $\pi_N$ ...
2
votes
1answer
38 views

Cantor's diagonal argument modified version

I have the following doubt regarding Cantor's diagonal argument. First of all, the "usual case" is quite clear for me. If $X$ is some set, then we can show there is no surjection from $X$ onto the set ...
0
votes
1answer
17 views

$f_n(x) = \left\lfloor \frac{\sin(2\pi (x / n + 1/ 4) + 1 }{2}\right\rfloor$ and related

$f_n(x) = \left\lfloor \frac{ \sin(2\pi (\frac{x}{n} + \frac{1}{4})) + 1}{2}\right \rfloor = 1 \iff x = kn$ and $ f_n(x) = 0 \iff x \neq kn$. Let $g_n(x)$ be what's within the floor brackets. Then ...
1
vote
0answers
35 views

Least squares solutions of the linear system

I'm doing problems from old exams, and my solutions don't add up with the professor's solution. The problem is as followed: Find all least squares solutions of the linear system. I checked my ...
1
vote
2answers
42 views

Prove that no set can contain everything (or every other set)

Prove that there cannot exist a set that contains everything. Ill put my proof in the answer so please check it there. Also if there is a more creative way to do this(using the basic axioms) if it's ...
0
votes
1answer
49 views

Binomial theorem proof

I'm working through Richard Hamming's "Methods of Mathematics Applied to Calculus, Probability, and Statistics" on my own. I'm struggling with his proof of the binomial theorem, as summarized below. ...
2
votes
1answer
25 views

A surjective endomorphism (of a Noetherian ring) is injective.

The problem is stated as follows: "Let $R$ be a Noetherian ring and $\theta$ be a ring homomorphism from $R$ to $R$. Show that if $\theta$ is surjective then it is also injective." Regardless of the ...
1
vote
3answers
62 views

Combinatorial Proof of a Simple Identity

Consider the following identity: $\binom n r = \frac n r \binom {n-1} {r-1}$ where $n \ge r \ge 1$. It's easy to supply an algebraic proof, but I'm looking for a combinatorial proof. I tried the ...
3
votes
1answer
36 views

If $b \equiv 0 \pmod a$ and $c \equiv 0 \pmod b$, then $c \equiv 0 \pmod a$

The question is If $b \equiv 0 \pmod a$ and $c \equiv 0 \pmod b$, then $c \equiv 0 \pmod a$. My attempt is that $b \equiv 0 \pmod a$ can be written $a\mid b-0 = a\mid b$ and the same with $c \equiv 0 ...
2
votes
1answer
30 views

Evans pde book: details on an bound for a Sobolev norm in the proof of the Meyers-Serrin theorem

Let $U$ be an open subset of $\mathbb{R}^n$ and $f\in W^{m,p}(U)$. Suppose that $$\|f\|_{W^{m,p}(V)}\leq\delta\tag{1}$$ for all $V\subset\subset U$ (that is, all $V$ such that $V\subset\overline{V} ...
2
votes
2answers
51 views

$\mathfrak{sl}(2)$ is a simple Lie algebra.

I am trying to prove that $\mathfrak{sl}(2,\Bbb C)$ is simple. Since this takes the $[x,y]=xy-yx$ matrix commutator bracket, this is clearly non-abelian. So to prove it is simple, we need only show ...
3
votes
2answers
75 views

Question about irrationality proof of $\sqrt{n}$

I'm talking specifically about a proof that I've found. I don't seem to get some parts of it. It states that if you take: $$\sqrt{n}=\frac{p}{q} \:\: \;\;p,q \in \mathbb{Z} $$ where $p$ and $q$ share ...
3
votes
2answers
37 views

Unique Linear Map- Linear Algebra

Let $E = {e_1, . . . , e_n}$ be a basis for $\mathbb{R}^n$ , and let $v_1, . . . , v_n$ be arbitrary vectors in $\mathbb{R}^m$. Prove that there is a unique linear map $T : \mathbb{R}^n \rightarrow ...
0
votes
1answer
40 views

Graphic proof of an inequality between sequence ratios

I would like to verify my proof for the following claim. Let $b_i$ be a positive decreasing sequence, $j<k$ two integers and $d$ a positive number. Prove that: $$ ...
0
votes
1answer
20 views

What am I doing wrong here? Showing $\text{Ord}_{N}(a)|k\iff a^k\equiv 1 \pmod N$.

Show $\text{Ord}_{N}(a)|k\iff a^k\equiv 1 \pmod N$ where $a$ is invertible. What I did is: If $\text{Ord}_{N}(a)|k$ it is obvious. Suppose $a^k\equiv 1 \pmod N$. Not let us assume by contradiction ...
1
vote
1answer
53 views

Limit of arithmetic means

If $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\dfrac{s_0+s_1+\dots+s_n}{n+1}.$$ If $\lim s_n=s$, prove that $\lim \sigma_n=s.$ My proof: Let $t_n:=s_n-s$ ...
0
votes
3answers
29 views

Power series with radius convergence $\leqslant 1$

Suppose that the coefficients of the power series $\sum a_nz^n$ are integers, infinitely many of which are distinct from zero. Prove that the radius of convergence is at most 1. Proof: Let radius of ...
1
vote
1answer
43 views

Proof that $A\cap\emptyset=\emptyset$

I'm trying to prove $A\cap\emptyset=\emptyset$. I've seen several proofs for this which all seemed to essentially go about proving it by noticing that $\emptyset\subset A\cap\emptyset$ by definition ...
1
vote
1answer
13 views

Proving with divisibility

I have never written any proofs (except high school geometry) in my life, so I'm not sure what exactly the proper formatting should be. Involving divisibility, the proposition states: Let $a, b,$ ...
1
vote
2answers
44 views

Proof about sum of convex polygon interior angles

I'm working through Richard Hamming's "Methods of Mathematics Applied to Calculus, Probability, and Statistics" on my own. I'm struggling with this proof (clipped from Google books): I follow him ...
0
votes
0answers
22 views

Let $R$ be a relation on $A$ and let $S$ be the transitive closure of $R$. Prove that $\text{Dom}(S) = \text{Dom}(R)$.

This is from "How To Prove It". The full exercise also asks to prove that $\text{Ran}(S) = \text{Ran}(R)$ but I was set from the outset on proving that $\text{Dom}(S) = \text{Dom}(R)$ first. Since the ...
0
votes
1answer
55 views

Proving well-ordering property of natural numbers without induction principle?

In Munkres, Topology, he has this way of proving the well ordering property for the natural numbers: He assumes he can work with the real numbers from the for the real numbers Then he defines an ...