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

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1answer
21 views

Is the supremum of a subset of functions element of the larger set?

I tried to prove the following statement and would like to know whether my proof contains errors. Let $X$ be the set of continuous functions $f: [0,1] \rightarrow [0,1]$ and let $\geq$ be a relation ...
2
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2answers
118 views

If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.

If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$. My proof: Let $|G| = 2n$. Since G is finite, there exists, $a \in G$ such that $a^p = e$ and by Lagrange's ...
2
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1answer
33 views

[Verification]Let H be a subgroup of G and $N = \bigcap_{x\in G} xHx^{-1}$. Prove that N is a subgroup of G and that $aNa^{-1} = N$ for all $a \in G$

Let $H$ be a subgroup of $G$ and $$N = \bigcap_{x\in G} xHx^{-1}$$. Prove that $N$ is a subgroup of $G$ and that $aNa^{-1} = N$ for all $a \in G$. Proof: $1 = x1x^{-1}$ for all $x \in G$ . Hence $1 ...
1
vote
1answer
16 views

[Proof Verification]If $u, v \in I$ satisfy $c-\delta < u \leq c \leq v < c+\delta$, then we have $f(v) -f(u) -(v-u)f'(c)| \leq \epsilon(v-u)$.

Let $f:I\to \mathbb{R} $ be differentiable at $c \in I$. Establish the Straddle Lemma: Given $\epsilon >0$, there exists $\delta >0$ such that if $u, v \in I$ satisfy $c-\delta < u \leq c ...
2
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0answers
70 views

If $X$ is finite and $R$ is a complete and reflexive binary relation on $X$, then $M(R, S) \neq \emptyset$ on any $S \subset X$ iff $R$ is acyclic.

Could you help me to verify my proof and my writing? Definition 1: A binary relation $R$ on $X$ is complete if, for all $x, y \in X$ such that $x \neq y$,$xRy$ or $yRx$ or both and reflexive if, for ...
2
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1answer
39 views

Using $x = y - b/2$ to solve a quadratic equation

I've been reading a book called Mathematics for the Nonmathematician, and it presents a solution to quadratic equations of the form: $x^2 + bx + c = 0$ which relies on coming up with a new formula, ...
2
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2answers
34 views

counter example of equicontinuous

Consider the functions on $[0,1]$: $f_n(x)=nx$, when $x$ is between $0$ and $1/n$ $f_n(x)=2-nx$, when $x$ is between $1/n$ and $2/n$ $f_n(x)=0$, otherwise How to see it is not (uniformly) ...
1
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1answer
42 views

What is wrong with the following argument involving Fibonacci and Lucas numbers?

The Lucas numbers $L_n$ are defined by the equations $L_1 = 1$, and $L_n = F_{n+1} + F_{n-1}$ for each $n \geq 2$. What is wrong with the following argument? Assuming $L_n = F_n$ for $n = ...
2
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0answers
56 views

Is this proof of a mathematical olympiad problem correct?

I'm quite sure about the exactness of my proof, but I'd like to hear (constructive) criticism about my writing. This is the problem: Every non-negative integer is coloured white or red, so that: 1) ...
2
votes
2answers
32 views

[Proof Verification]Prove that if f is differentiable at $c \in I$ and $f'(c) = 0$, then g is not differentiable at $d:=f(c)$.

Proposition. Let I be an interval, and let $f: I \to \mathbb{R}$ be a strictly monotone and continuous on I. Let $J := f(I)$ and let $g:J \to \mathbb{R}$ be the inverse function of f. Prove that if f ...
2
votes
1answer
12 views

Is a permutation of block diagonals similar?

Let $A=B_1\oplus\cdots \oplus B_n$. Let $\sigma\in S_n$ be a permutation. Then are $A$ and $B_{\sigma(1)}\oplus \cdots \oplus B_{\sigma(n)}$ similar? I have proven that this is true, but I want to ...
1
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1answer
74 views

“Prove that for any $n \times n$ matrix $A$ and $B$, $AB-BA \neq I$”

So this is a exercise from the course compendium for a matrix course I'm currently taking. "Prove that for any $n \times n$ matrix $A$ and $B$, $AB-BA \neq I$" Is the proof that I have constructed a ...
3
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2answers
175 views

A natural number $n>2$ is a prime iff $\prod_{k=1}^{n-1} k \equiv n-1 \pmod {\sum_{k=1}^{n-1} k}$

Is this proof acceptable ? Theorem 1 (Wilson). A natural number $n>1$ is a prime iff: $$(n-1)! \equiv -1 \pmod n.$$ Theorem 2. A natural number $n>2$ is a prime iff: ...
1
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0answers
34 views

Proof of the Box-Muller method

This is Exercise 2.2.2 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise (Box–Muller method): Let $U$ and $V$ be independent random variables that are uniformly ...
2
votes
3answers
42 views

Proof of divisibility, given divisibility of a square

The below proof is incorrect. See the answers for more information. This question is in the context of exploring how to explain the process of developing a proof. When reading a proof on the ...
2
votes
1answer
41 views

Proof Verification of $B \cup A = B$ implies $Pr(A) \leq Pr(B)$

Basic Information If you're confused $Pr(A)$ stand for probability of A. My Work 1) $A\cup B = B \iff A \subseteq B$ (By Theorem 3.4 in our textbook) 2) $A \subseteq B \implies |A| \leq |B|$ ...
1
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2answers
31 views

My proof of: $|x - y| < \varepsilon \Leftrightarrow y - \varepsilon < x < y + \varepsilon$

Is it reasonable to prove the following (trivial) theorem? If yes, is there a better way to do it? Let $x, y \in \mathbb{R}$. Let $\varepsilon \in \mathbb{R}$ with $\varepsilon > 0$. ...
0
votes
1answer
59 views

Attempt to proof the Cantor-Bernstein theorem

I've found a proof of the Cantor-Bernstein theorem in Kleene's 'Introduction to Metamathematics' (1952) in §4 Thm A. I must admit I don't understand its essence but I was wondering if the proof could ...
3
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3answers
76 views

Is there a proof for the fact that, if you perform the same operation on both sides of an equality, then the equality holds?

Is there a proof for this, or is it just taken for granted? Does one need to prove it for every separate case (multiplication, addition, etc.), or only when you are operating with different elements ...
1
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1answer
25 views

My proof of: Given an adherent point, some sequence converges to it.

(I have two specific questions.) Is my proof correct? Are style, wording, and punctuation alright? $\textbf{Definition 1.}$ Let $X \subseteq \mathbb{R}$. Let $x \in \mathbb{R}$. The point $x$ ...
1
vote
1answer
18 views

[Verification]$G$ is a group whereby $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show $G$ is abelian.

If $G$ is a group in which $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show that $G$ is abelian. Proof: Let $x$ be the smallest of the 3 consecutive ...
1
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0answers
15 views

Understanding a proof regarding T-annihilators

This may be a silly question but I don´t undersand one part of the next proof: Let $T:V\to V$ be a cyclic linear operator on a finite dimensional vector space with minimal polynomial $p^k(x)$ ($p$ is ...
4
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3answers
67 views

The quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open.

I'd like to show that the quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open, where I'm considering $\mathbb P^n$ as the quotient space of $\mathbb R^{n+1} \setminus \{0\}$ ...
0
votes
1answer
31 views

Proof Using iff Intermediate Lines

I am posting this question motivated by Bungo's response to my question here -- scroll down to his/her response and comment. It was the first time I've seen this technique. It looks like a circular ...
0
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1answer
12 views

$\{h\in A^B|h \text{ is invertible}\}$ is equiumerous to $\{k\in B^A|k \text{ is invertible}\}$ and $\aleph_0$ right invertibles for a function

1.Let $A,B$ be sets, prove: $\{h\in A^B|h \text{ is invertible}\}$ is equinumerous to $\{k\in B^A|k \text{ is invertible}\}$ 2.Let $A,B$ be sets and a function $f\in A^B$ give an example right ...
1
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1answer
23 views

Locally Compact Spaces: Separation Property

Given a locally compact Hausdorff space. Every compact set has a compact neighborhood base: $$C\subseteq U:\quad N\subseteq U\quad(C\subseteq N^°)$$ My construction was contrary to Rudin's: ...
0
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0answers
28 views

Does there exist a basis $(p_0,p_1,p_2,p_3)\in P_3(\Bbb F)$ such that none of the polynomials $p_0,p_1,p_2,p_3$ has degree $2$?

Does there exist a basis $(p_0,p_1,p_2,p_3)\in P_3(\Bbb F)$ such that none of the polynomials $p_0,p_1,p_2,p_3$ has degree $2$? First thing is trying to understand what is being asked of me. Is ...
1
vote
1answer
25 views

Next step to take in direct proof for one to one?

This is from Discrete Mathematics and its Applications And the definition of strictly increasing. Here is my work so far. I know that a direct proof involves making an assumption p, which in ...
10
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4answers
376 views

How can I complete this proof by contradiction?

This problem is from Discrete Mathematics and its Applications: Prove that there are no solutions in integers $x$ and $y$ to the equation $2x^2 + 5y^2 = 14$. I am trying to use proof by ...
0
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1answer
59 views

Kleene star proof: $(AA)^*$ is the set of strings from $A^*$ of even length

Let $A$ be an alphabet. By $A^{**}$ let us denote the set of all strings from $A^*$ of even length. (This definition may be incorrect but it was given to me in the question) Show that ...
1
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1answer
86 views

Cardinality of set of all bijections $\mathbb{N}\to\mathbb{N}$; is my proof correct?

I need to find cardinality of a set containing all bijections $\mathbb{N} \to \mathbb{N}$. My proof goes like that: Let $S$ be the set containing all bijections $\mathbb{N} \to \mathbb{N}$. There ...
0
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0answers
25 views

Let $P$ with real coefficients satisfies $|P(i)|<1$. Prove that there is a root $z=a+bi$ of $P$ such that $(a^2 + b^2 + 1)^2 < 4b^2 + 1$

A monic polynomial $P$ with real coefficients satisfies $|P(i)|<1$. Prove that there is a root $z=a+bi$ of $P$ such that $(a^2 + b^2 + 1)^2 < 4b^2 + 1$ One solution is: Let us write $P(x) = ...
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4answers
48 views

A Proof that Orthogonal Complement is unique

So our professor asked us to prove that considering any subspace $S$ of a vector space $V$, the orthogonal complement $S^{\perp}$ is unique. I have devised a proof and I am not sure whether this ...
2
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2answers
72 views

Proving by induction that any two natural numbers are equal.

This is something I've been working on for a while now; although it seems trivial, I am confused. I can't seem to find the error. Originally I thought the problem was with the base case, then I ...
2
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1answer
36 views

Disprove that if $f([-2, 2])$ is an interval, then $f$ is continuous

Disprove that if $f([-2, 2])$ is an interval, then $f$ is continuous My counter example is $\begin{cases} 1 - x & \text{ if } -2 \leq x \leq 1 \\ 2 - x & \text{ if } 1 < x \leq 2 ...
2
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2answers
30 views

Next step to take to reach the contradiction?

This problem is from Discrete Math and its Applications I am trying to use proof by contradiction to do this problem, proof by contradiction as described by the book Here is my work so far for ...
0
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2answers
44 views

How to prove that the Nested Interval Theorem fails to hold in $\mathbb Q$?

Claim: The Nested Interval Theorem does not hold in $\mathbb Q$. I can prove this by using sequences $a_n$ and $b_n$ where $a_n < b_n$ and they both converge for an $x$ which is any irrational ...
0
votes
1answer
15 views

Finding a bijective function from $\prod_{i\in I}X_i$ to $\bigl(\prod_{j\in J}X_j\bigr)\times\bigl(\prod_{k\in K}X_k\bigr)$

If $(X_i)_{i\in I}$ is a family of sets and $J,K$ are non-empty disjoint sets of $I$ such that $I=J\cup K$, then show that there is a bijective function from $\prod_{i\in I}X_i$ to ...
1
vote
1answer
76 views

Surjection $f$ induces surjection $\mathcal P (f)$ on power sets

Just wanted to make sure the way I approach this was correct because it seemed a bit too simple for an answer: Question: Let $\mathcal P \left({X}\right)$ represent the power set of $X$. Let $f: X ...
4
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0answers
36 views

My proof of: Every convergent real sequence is a Cauchy sequence.

Is my proof correct? Let $(x_n)_{ n \in \mathbb{N} }$ be a real sequence. $\textbf{Definition 1.}$ $(x_n)$ is $\textit{convergent}$ iff there is an $x \in \mathbb{R}$ such that, for every ...
0
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2answers
28 views

Mathematical induction (prove divisibility)

My attempt at the solution is to let P(n) be $10^{3n} + 13^{n+1}$ P(1)= $10^3 + 13^2 = 1169$ Thus P(1) is true. Suppose P(k) is true for all $k \in N$ $\Rightarrow P(k) = 10^{3k} + 13^{k+1} = ...
0
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4answers
98 views

Why can't I prove this statement by simple induction? Sum of $1/2^1 + \cdots+ n/2^n = x$

I have to prove the following: $$ \frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n}=2-\frac{2 + n}{2^n}. $$ I am trying to prove this by simple induction. First, I proved that $P(1)$ ...
2
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1answer
52 views

Can the average of an unbounded sequence of positive numbers be 0?

Looking for verification of a proof. Find $\{s_n\}$ a sequence of positive numbers such that $\limsup_{n\to\infty}s_n=\infty$, and $\lim_{n\to\infty}\sigma_n=0$ where ...
3
votes
1answer
130 views

On a proof of Riesz-Fischer Theorem

Questions : [See below for context.] $\rm\color{#c00}{a)}$ First, is the proof presented below $100$ % correct ? $\rm\color{#c00}{b)}$ How would one justify the LHS of $(2)$ ? Are my ...
0
votes
1answer
25 views

How to introduce bi-conditional in this proof?

This is from Discrete Mathematics and its Applications Just for context, I know that the universal set is everything and that the complement of A is just difference of the universal set and A. A ...
0
votes
1answer
19 views

Distance to a closed ball in a normed space.

Let $(E, \|\cdot\|)$ be a normed vector space, and consider $B = B[{\bf a},r]$ the closed ball. Let ${\bf b}\in E$. Then $\newcommand{\d}{{\rm d}} \d({\bf b},B) = 0$ if and only if ${\bf b} \in B$. ...
1
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0answers
26 views

Finding out if a function is invertible: $f,g:\mathbb N\to \mathbb N$, $g(x)=2x$ and $f$ with cases

Let $f,g:\mathbb N\to \mathbb N$ such that $g(x)=2x$ and $f(x)=\begin{cases}\frac x 2 &, x\in\mathbb N_{even}\\ x+9 &,x\in\mathbb N_{odd}\end{cases}$ ...
3
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0answers
21 views

Proof Verification: Let S be a non-empty subset of a ring R. Then S is a subring of R if and only if S is closed under $-$ and $\times$.

Let S be a non-empty subset of a ring R. Then S is a subring of R if and only if S is closed under $-$ and $\times$. Proof: First, prove that S is a subgroup of R. Pick an arbitrary element $x$ from ...
1
vote
1answer
25 views

Prove subset of $\mathbb{C}^n$ is convex and complete

I have to prove that the subset $M=\sum_{i=1}^n x_i=1$ of $\mathbb{C}^n$ is convex and complete w.r.t. the inner product $<x,y>=\sum_{i=1}^n x_i\bar{y_i}$. Now being convex is trivial. However ...
1
vote
0answers
51 views

Why this proof is incorrect?

I have an exercise that I cannot really understand: Let P(n) be a property over the naturals (i.e., $n \in N$). The induction axiom, taking $0$ for the base-case instance, is the formula: $$[P ...