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

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1
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1answer
20 views

Is this divisibility proof by induction correct/sufficient?

To show: 13 | $4^{2n+1}+3^{n+2}$ I used induction beginning successfully with n=0 (or n=1), then making the step to n+1: An x exists so that $13x = 4^{2n+3}+3^{n+3}$ $13x = 16*4^{2n+1}+3*3^{n+2}$ ...
-1
votes
2answers
70 views

Quotient-Remainder Theorem Proving [closed]

This theorem is obviously correct. Now I try to prove it by well-ordering principle. But I don't know where to start the proving....
0
votes
1answer
24 views

Proof: $a,b \in \overline{\Bbb{R}}$, $a\leq b < +\infty \wedge a \in \Bbb{R} \Rightarrow b \in \Bbb{R}$

I must to proof the following: Prop.: let be $a,b \in \overline{\Bbb{R}}$ then:$$a\leq b < +\infty \wedge a \in \Bbb{R} \Rightarrow b \in \Bbb{R}$$ Proof: by contradiction I have $b \notin ...
5
votes
0answers
80 views

$f$ continuous nowhere implies $f$ not Riemann integrable

I am trying to prove the statement in the title, without using the theorem which says that the set of discontinuities of a Riemann integrable function must have measure $0$. I am aware that similar ...
1
vote
1answer
36 views

Why doesn't a linearly independent set of image vectors imply an injection?

While researching a question I had, I came across this post. Without reading the answers, I started working on it myself and eventually came up with a proof for this statement: Let $f \in ...
1
vote
1answer
44 views

Explicit proof of the derivative of a matrix logarithm

Firstly, I'm but a mere physicist, so please be gentle :-) I want to explicitly show that the derivative of the (natural) logaritm of a general $n \times n$ (diagonalizable) matrix $X(x)$ w.r.t. $x$ ...
0
votes
1answer
44 views

Isometry is not surjective

According to the definition I am using, an isometry is a mapping $f:X \rightarrow Y$ between two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$: $$ d_{Y}(f(a),f(b)) = d_{X}(a,b) $$ for all $a,b \in X $ I ...
2
votes
2answers
69 views

Prove P∧(Q∨R)→(P∧Q)∨(P∧R) from P∧(Q∨R) using Natural Deduction

I am trying to prove P∧(Q∨R)→(P∧Q)∨(P∧R) from P∧(Q∨R) using Natural Deduction. Here is my attempt using JAPE application. ...
0
votes
2answers
27 views

Infinimum of a set

Given the following conditions: $x \in \Bbb R$ and $y\in (0,1)$ I was asked to prove that inf $ |x-y |=0 $ My Attempt: By the elementary properties of the modulus function , we know that $ 0 ...
2
votes
1answer
50 views

Metric Spaces: The dist function

Given that $A$ is defined as non-empty subset of $(X,d)$ The distance function is defined as such: $dist(x,A)=$ inf $_{y\in A} \lbrace d(x,y) \rbrace $ Given the above we are asked to prove the ...
1
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2answers
33 views

Show the equivalence: $ab|c \iff a|c$ and $b|c$

Let $a,b \in \mathbb{Z}$ \ {$0$} with $gcd(a, b) = 1$ and let $c \in \mathbb{Z}$. Show the equivalence: $ab|c \iff a|c$ and $b|c$ Also give an example of numbers $a,b \in \mathbb{Z}$ \ {$0$} and ...
0
votes
1answer
26 views

Set proof (symmetric difference of disjoint set)

The question: Prove this is true: ($A$ $\setminus$ $B$) $\cup $ ($B$ $\setminus$ $A$) = ($A$ $\cup$ $B$) iff ($A$ $\cap$ $B$) = $ \emptyset$ ...
2
votes
2answers
36 views

Proof that matrix $B^{-1}$ = matrix $A^{-1}$ with 2 columns swapped given that B = A with 2 rows swapped.

I'm trying to prove the following. Given that $A$ is a nonsingular $n \times n$ matrix, and $B$ is the nonsingular matrix obtained by interchanging rows $i$ and $j$ of $A$, where $i \neq j$, show ...
4
votes
1answer
36 views

Unique representation of a degenerate simplex

I recently learned about simplicial sets, and now I'm studying some basic properties. I've learned that every degenerate simplex is a degeneracy of a unique non-degenerate (nice) simplex. However, I ...
4
votes
1answer
66 views

Show that $f(z)=\sum_{n= 0}^{+\infty}a_n z^n$ is a polynomial

Let $f(z)=\sum_{n= 0}^{+\infty}a_n z^n$, the radius of convergence $\ge 1$. For all $n,\quad a_n\in \mathbb{Z}$ and $f$ is bounded the open unit disk. Show that $f$ is a polynomial. My ...
5
votes
2answers
45 views

Proving well definedness of addition in real numbers constrructed from cauchy sequences.

While studying real analysis, I got confused on the following issue. Suppose we construct real numbers as equivalence classes of cauchy sequences. Let $x = (a_n)$ and $y= (b_n)$ be two cauchy ...
0
votes
0answers
34 views

Proof verification problem

I would like to know what is the true output, and what is the way of solving it? To me, I have got solution to be exact as Q1.
2
votes
3answers
58 views

Prove $\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$

I need to prove the following: If $n,m,k\in \mathbb{N}$ and $k\leq m \leq n$, then $$\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$$. I did the following steps: \begin{align} ...
-1
votes
0answers
41 views

What is wrong with this induction proof for a closed form recursive function?

I need some help, I can't seem to find What is wrong in this proof, yet I'm not getting what I need. Anyone know? Prove by induction $2(S(2^{k-1})) + 2^k = 2(2^{k-1})k +2^k$, given that: $S(2^0 ) = ...
0
votes
2answers
31 views

Anyone have a good proof for the second part of FTC?

Does anyone have a good proof for the second part of the fundamental theorem of calculus? I haven't been able to find any good videos on it so far so I'd like someone to write it down and I can throw ...
0
votes
1answer
26 views

Images of basis vectors under injective linear map form a linearly independent set

I missed this question on a quiz: Prove that if $\{v_1, \ldots v_n \}$ is a basis for $V$ and $f\,:\,V\rightarrow W$ is an injective linear map, then $\{f(v_1), \ldots f(v_n)\}$ is linearly ...
3
votes
1answer
36 views

Identification of $L^2$ limits with distributional convergence

I just read the thread on "too much effort" and I would like to be more specific. Is the following reasoning correct: Let $g,g_\delta\in H^1(D)$, $D$ some domain in $\mathbb{R}^n$ with the following ...
0
votes
4answers
52 views

Prove $(A \cap B) \cup (A \cap B^c) = A$

I need to prove the following statement: $$(A \cup B) \cap (A \cup B^c) = A$$ I did the following steps: \begin{align} &(A \cup B) \cap (A \cup B^c) = A \\ &A \cup B \cap (A \cup A) = A \\ ...
2
votes
3answers
45 views

How can I prove $2^n > n^2 $ by induction using a basis $> 4$ [duplicate]

I've been trying to prove this statement by induction; however, in following the steps I normally take I end up utterly stuck. I know that I must be missing something, but I have been stuck on this ...
1
vote
1answer
62 views

Possible book correction or am I missing something?

Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y ...
0
votes
1answer
28 views

The proportion of $\omega$s in $A$ converges almost surely to $P(A)$

Let $A$ be an event in $(\Omega,\mathcal{F},P)$. We generate independent inquiries from $\Omega$ in accordance to $P$. Show that the proportion of $\omega$s in $A$ converges almost surely to ...
20
votes
5answers
3k views

Is a brute force method considered a proof?

Say we have some finite set, and some theory about a set, say "All elements of the finite set $X$ satisfy condition $Y$". If we let a computer check every single member of $X$ and conclude that the ...
0
votes
0answers
25 views

Linear Programming Problem(Algebra of Simplex Method)

May I know if my proofs to the following claims are correct? Please advise. Thank you. 1.) Reduced cost corresponding to basic variables are zero. Proof: Consider the standard L.P. : max ...
0
votes
0answers
43 views

Describe the kernel and the fibers of $\phi$ geometrically (as subsets of the plane).

Define $\phi : \mathbb{C}^{\times} \mapsto \mathbb{R}^{\times}$ by $\phi(a+bi) = a^2 + b^2$. Prove that $\phi$ is a homomorphism and find the image of $\phi$. Describe the kernel and the fibers of ...
1
vote
1answer
28 views

Direct sum of $3$ subspaces

$V_1$,$V_2$,$V_3$ are subspaces of vector space $V$. How to prove that if $V_1 \cap \left(V_2+V_3\right) = V_2 \cap \left(V_1+V_3\right) = V_3 \cap \left(V_2+V_3\right)=\{0\}$ so $V_1\oplus V_2 ...
1
vote
1answer
47 views

Orthogonal matrices, their determinant and eigenvalues

Once again! Let What to do? Find eigenvalues for $A, B, AB, BA$. How I want to do this: $A = \begin{pmatrix} cos(a) & -sin(a) & 0 \\ sin(a) & cos(a) & 0 \\ 0 & 0 & 1 ...
0
votes
1answer
51 views

Calculating the first order partial derivatives of the Gaussian function

I am trying to calculate the first order partial derivatives of the Gaussian function. My calculations look correct to me but when I implement them in a C program I do not get the desired result. So, ...
0
votes
2answers
29 views

Proof $ GCD(a,b) = GCD(a, b-a) = GCD (a, r_b) $

let $a,b \in \mathbb{N}$ and a < b. let $r_b$ the rest when dividing b through a. (1) If $r_b$ is the rest, then there exists a q so that $ b = q*a + r_b $. (2) Now I show: $gcd(a,b) = gcd(a, ...
1
vote
2answers
82 views

Proof Verification - Every sequence in $\Bbb R$ contains a monotone sub-sequence

Came across the following exercise in Bartle's Elements of Real Analysis. This is the solution I came up with. Would be grateful if someone could verify it for me and maybe suggest better/alternate ...
0
votes
1answer
39 views

Clarification of the proof of a theorem: Triangle inequality

First: My proof of the triangle inequality: If $a,b \in \mathbb{R}$, then $|a+b| \leq |a| + |b|$ Proof: Consider the 4 cases: 1) $a<0$ and $b<0$ 2) $a>0$ and $b<0$ 3) $a>0$ ...
2
votes
2answers
97 views
2
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3answers
53 views

Proving that $\lambda$ being an eigenvalue for $A$ implies $\lambda^{-1}$ is an eigenvalue for $A^{-1}$

Let $A$ be an invertible matrix, and let $\lambda$ be an eigenvalue for $A$. We have that $Ax = \lambda x$ for some eigenvector $x$. Note that $A^{-1}Ax = A^{-1}\lambda x$, which gives $x = ...
1
vote
1answer
67 views

Proof that $ [2x] + [2y] \ge [x]+ [x+y] + [y]$?

I have to proof (or disprove) the following $ [2x] + [2y] \ge [x]+ [x+y] + [y]$ for $x,y \in \mathbb{R}$. [x] and [y] means the floor-function. Can I do the following? (1) Assume $x,y \in [0,1]$, ...
0
votes
1answer
36 views

Showing that $\log \log(z)$ is Analytic (Proof Verification)

Goal: Convert $\log \log (z)$ into a single-valued function defined on a suitable region of $\mathbb{C}$ and then prove that it is analytic. Attempt: As has been demonstrated elsewhere, we have ...
1
vote
1answer
15 views

Weak monotonicity of ordinal addition

I'm trying to prove the weak monotonicity of ordinal addition, i.e. if $\alpha \leq \beta$, then $\alpha + \gamma \leq \beta + \gamma$. The proof is not all that difficult, but I want to make sure I ...
0
votes
0answers
49 views

Showing that $\sqrt{1+z} + \sqrt{1-z}$ is Analytic (Proof Verification)

Ahlfors: Give a precise definition of a single-valued branch of the function $\sqrt{1+z} + \sqrt{1-z}$ in a suitable region, and prove that it is analytic. Is my following proof attempt valid? ...
2
votes
1answer
77 views

Proof Verification: Show sequence is bounded and find limit: $x_1 \gt 1$ and $x_{n + 1} = 2 - \frac{1}{x_n}$

Came across the following exercise in Bartle's Elements of Real Analysis and am a little unsure about my solution. Would be extremely grateful if someone could verify it for me. Let $x_1 \in ...
0
votes
1answer
32 views

One-to-one and Onto: True or False

1) Suppose ${f}: X\to Y$ is one-to-one and $A\subseteq X$. Then $f^{-1}({f}(A))=A$. True 2) Suppose ${f}: X \to X$, and assume that ${f} \circ {f}$ is one-to-one and onto. Then ${f}$ is one-to-one ...
2
votes
1answer
42 views

contracting an associated prime in a local ring homomorphism

Let $\phi: (R,m) \rightarrow (S,n)$ be a morphism of local Noetherian rings. Let $M$ be an $S$-module which is finitely generated as $R$-module. Let $p \in \operatorname{Ass}_S M$ and let $x \in M$ be ...
7
votes
5answers
288 views

If $\mathcal{P}(A)=\mathcal{P}(B)$, then $A=B$? [duplicate]

Prove, disprove, or give a counterexample: If $\mathcal{P}(A)=\mathcal{P}(B)$, then $A=B$. Assume $\mathcal{P}(A)=\mathcal{P}(B)$. Since we know $A \subseteq A$, we know $A \in ...
3
votes
1answer
47 views

Linear Algebra: Identity map

I was asked to prove that the identity map $id : \Bbb R^n \to \Bbb R^n $ can be represented by the the identity matrix regardless of the basis My Attempt: Let $\mathcal B = \lbrace v_1 , ...,v_n ...
3
votes
0answers
53 views

Proof about sequences of functions.

Is this proof correct? If $\{f_{n}\}$ is a sequence of functions in $C(X,Y)$, $X$ compact, $Y$ complete, and the sequence converges, to $f$, then $K=(\bigcup\{f_n\})\cup \{f\}$ is closed. Proof. ...
0
votes
0answers
57 views

A question on odd perfect numbers

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(M) = 2M$, then $M$ is said to be perfect. Currently, there are $48$ known examples of even perfect numbers -- on ...
1
vote
2answers
61 views

$\forall x \in \mathbb{R}$ show that $x=\sum_{n=1}^\infty k_na_n = \prod_{n=1}^{\infty}m_na_n$ …

Yet again, another cool problem from the book "problems in mathematical analysis" by Piotr & Witkowski: Prove that if $a_n \neq 0$, $n=1,2,\cdots$ and $\displaystyle \lim_{n \to \infty} a_n = 0$, ...
4
votes
1answer
39 views

Problems with a proof that -in a linear order- a minimal element is the smallest element

I have a problem with a proof I found in Velleman's "How to prove it". This is sort of interesting, because it is the very first time I cannot see the structure of a proof presented in the book. The ...