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

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Exercise 3.3.8 from Understanding Analysis by Stephen Abbott

Motivation: trying to prove that if $K \subseteq \mathbb{R}$ is compact (and thus, by the Heine-Borel theorem, closed and bounded), then this implies that any open cover for $K$ has a finite subcover. ...
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1answer
13 views

Proof check - equivalence classes are intervals

We define $\sim$ for a nonempty subset $X\subseteq\mathbb{R}$ by: $x\sim y$ if $\lbrack\min\lbrace x,y\rbrace,\max\lbrace x,y\rbrace\rbrack\subseteq X.$ This is an equivalence relation on $X.$ I want ...
3
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1answer
118 views

Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element. ...
3
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0answers
27 views

$\sin(nx)$ does not contain Cauchy subsequence in $L^p([0,2\pi]) $ for $1\leq p < \infty$

$\sin(nx)$ does not contain Cauchy subsequence in $L^p([0,2\pi]) $ for $1\leq p < \infty$ My attempt: Set $f_n(x) = \sin(nx)$. Argue by contradiction, suppose there exists a Cauchy ...
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1answer
33 views

Let m ∈ N. Define the relation ≡^ on Z by a ≡^ b for a, b ∈ Z if and only if a ≡ ±b (mod m).

(In other words, the relation ≡^ holds if either a ≡ b (mod m) or a ≡ −b (mod m).) Prove that the relation ≡^ on Z is transitive. ======= I believe there are 3 properties that it must meet ...
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1answer
54 views

every Abelian group is a converse lagrange theorem group

Let $G$ be a finite abelian group, then $G$ has a subgroup of order $n$ if and only if $n\mid G$. Proof: by Lagrange if $H\leq G$ then $|H|$ divides $|G|$ so this proves one of the implications. We ...
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1answer
28 views

Is my proof ok? Let $m \in \mathbb{N}$. Prove that the congruence modulo $m$ relation on $\mathbb{Z}$ is transitive.

Let $m \in \mathbb{N}$. Prove that the congruence modulo $m$ relation on $\mathbb{Z}$ is transitive. If $A$ is congruent to $B$ mod $m$ then $A - B = k m~~$ (1) If $B$ is congruent to $C$ mod ...
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1answer
49 views

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$. Proof: suppose $T:L^1 \rightarrow L^\infty$ continuous and onto. $L^1$ is separable, let $\{f_n\}$ be a countable dense ...
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1answer
47 views

Degree of maps $\mathbf{CP}^n\rightarrow\mathbf{CP}^n$

Theorem: Every map $f:\mathbf{CP}^n\rightarrow\mathbf{CP}^n$ has degree $k^n$ for some $k\in\mathbf{Z}$. Proof: Something which I don't understand is $\alpha^n\cap ...
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1answer
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Verify my proof: If $X$ is infinite, then there exists $f: \mathbb{N} \rightarrow X$ such that $f$ is injective.

Proposition: If $ X $ is infinite, then there exists $ f: \mathbb{N} \rightarrow X $ such that $f$ is injective. Proof: Define $X$ as a infinite set, i. e., there does not exist $ g: [k] \rightarrow ...
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1answer
42 views

Groups - Prove that every element equals inverse of inverse of element

This is my first proof about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. We use $^{-1}$ to denote inverse ...
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2answers
41 views

For random variables, show that $\limsup\limits_nX_n<\infty\Longrightarrow \sup_n X_n<\infty$

Why is the following true ? $$\limsup\limits_nX_n<\infty\Longrightarrow \sup_n X_n<\infty$$ where, $X_n's$ are random variables. If we consider only finitely many $X_n$, say ...
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2answers
35 views

Proof that the congruence relation on $\mathbb Z$ is transitive (attempt shown)

I have answered this question to the best of my knowledge but somehow I feel as if I am missing something? Can I further prove this statement or add anything to it? Question: Let $m \in \mathbb ...
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0answers
45 views

prove that $\sum_{k=1}^\infty|x_k y_k|$ converges

Let $V$ be the space of real sequences $x_k$ so that $\sum_{k=1}^\infty x_k^2$ converges. Let $\langle x,y\rangle=\sum_{k=1}^\infty x_k y_k$ Prove that $\sum_{k=1}^\infty |x_k y_k|$ converges My ...
2
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1answer
25 views

Multiplication between a normal subgroup and an arbitrary subgroup.

Given $G$ a group. $N$ a normal subgroup of $G$, and $H$ an arbitrary subgroup of $G$. Prove that $G=NH$ is a subgroup of $G$. I have to prove that $NH=HN$. But for every $h\in H$ we have that ...
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0answers
26 views

Proof verification: Munkres exercise 10, section 152

Can someone please verify my proof or offer suggestions for improvement? I'm thoroughly confused by this question, and I'm sure there's a mistake somewhere in my proof. Let $\{X_\alpha\}_{\alpha ...
3
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2answers
32 views

The topology generated by a basis is the intersection of all topologies containing that basis.

This question is from Munkres' Topology, section 13, exercise 5. I ask for verification and/or comments upon mistakes and inaccuracies. Let $\mathcal{A}$ be a basis for a topology on $X$. We are ...
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1answer
19 views

The intersection of a connected subspace with the boundary of another subset

Can someone please verify my proof or offer suggestions for improvement? Definition/Notation: The boundary of $A$, denoted by $\operatorname{Bd}(A)$, equals $\overline{A} \cap \overline{X-A}$. ...
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1answer
26 views

Suppose $R \sim_\omega R'$. Then for every $k$-tuple $a$ in $E$ and every natural number $p$, there is a $k$-tuple $b$ in $E'$ such that $a \sim_p b$

Sorry to bother you guys again with a Poizat question, but I'm struggling a little bit with the material (as it must be obvious) and I want to check if I got the main idea correctly or if I'm totally ...
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5answers
158 views

The set of all finite subsets of the natural numbers is countable

Could someone verify my proofs? Proposition: the set of all finite subsets of $\mathbb{N}$ is countable Proof 1: Define a set $ X=\{A\subseteq\mathbb{N}\mid \text{$A$ is finite} \}$. We can have a ...
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1answer
78 views

Continuous linear image of closed, bounded, and convex set of a Hilbert Space is compact

Is my proof of this proposition correct ? And is this proposition well known? Proposition: Let $C$ be a closed, bounded, and convex set in a separable Hilbert space $H$. Let $L : H \to \mathbb{R}^n$ ...
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2answers
40 views

Does A linear Transformation Of This Kind Exist?

Is there a linear transformation $T:\mathbb{R}_2[x] \rightarrow \mathbb{R}^3$ So that $Ker(T)=span \{ 1+x-x^2, 2+3x^2 \}$ $Im(T)=span \{ (0,0,9) \}$ What I have done: $Ker(T)=(\alpha+\alpha ...
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0answers
49 views

Prove that $T^{4} -12T^{2} +64$ is irreducible in $\mathbb{Q}[T]$

Is the following correct? I choose $3$ irreducible in $\mathbb{Z}$. If $g=(1+(3))T^{4} - (12 + (3))T^{2} + (64 + (3)) \in \mathbb{Z[T]}/(3)$ is irreducible, then $f=T^{4} -12T^{2} +64 \in ...
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3answers
120 views

Proof that the set of all functions from $\mathbb N$ to $\mathbb N$ is not enumerable

I'm trying to show that the set of all functions from $\mathbb N$ to $\mathbb N$ is not enumerable. Can someone verify my proof below? Proof: Let $\mathcal{F}(\mathbb{N}; \mathbb{N})$ be the set of ...
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3answers
67 views

How can I prove that $a^n + b$ is composite?

I need to know how could I prove that $2^{33} + 1$ is composite. Thanks!
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1answer
13 views

Proof Verification, any partially ordered set contains a maximal totally unordered subset (antichain).

I wrote a simple proof and I want to verify it. I simply argued that the set of all totally unordered subsets forms a partially ordered set, ordered by inclusion. Every chain in this partially ...
2
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1answer
47 views

Topology, Proof of function being continuous

Let $ (X_i,d_i),(Y_i,d_i^*)$, $i=1,\ldots,n $ be metric spaces. Let $ f_i:X_i \to Y_i, i=1,...,n $ be continuous functions. Let $$ X = \prod_{i=1}^{n} X_i , Y = \prod_{i=1}^{n} Y_i $$ and ...
0
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1answer
45 views

Using Borel-Cantelli Lemma

Let $X_1, X_2,\ldots$ be iid Geometric(p) where $p \in (0,1)$. Thus if $q=1-p$, then $P(X_n > k) = q^k$ for $k\geq 0$. Prove that for any fixed $\epsilon \in (0,1)$, $P(X_n > ...
6
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0answers
89 views

Proposed proof of analysis result

Hi please advise on my proof of the following result: Assume that $I \subset \mathbb{R}^{n}$ is convex, bounded open set with Lipschitz boundary and let $u_{m},u$ be such that $$u_{m} ...
1
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1answer
31 views

Adjoint of a compact operator between Hilbert space is compact

Let's $H$ be a Hilbert space and $T:H \rightarrow H$ a compact map. Show that the adjoint operator $T^*$ is also compact. I've been thinking a lot about this problem and this is what I've done: It's ...
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3answers
62 views

What is wrong with the proof of this incorrect formula for the probability of $A\cap B$?

The probability that events $A$ or $B$ (or possibly both at once) will occur is $P(A\cup B)$. Since we can think of $A\cup B$ as the set-theoretic analog of inclusive logical disjunction $\vee$, I ...
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1answer
51 views

What is wrong with the following induction argument?

I found a problem on a note on induction. The problem went like this: "Let $n$ be a non-negative integer. Suppose we are given a triangle and n points inside it, with no three of the given $n + 3$ ...
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1answer
39 views

Local minimum of the function:

Find the local minimum of the function: $$\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}$$ $$\f=\1^2-2\1\2+2\2^2+\3^2 \text{ in } \mathbb{R}^3$$ $\n\f=(2\1-2\2,-2\1+4\2,2\3) ...
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1answer
52 views

Proof on $F(x)=\sum_{n=1}^{\infty}f\left(\frac{x}{n}\right)$ uniform convergence and differentiability

Let $f$ be a function of $C^{\infty}$ class, such that $f(0)=0=f'(0)$. Prove that if $x\in\mathbb{R}$ and $$F(x)=\sum_{n=1}^{\infty}f\left(\frac{x}{n}\right)\ ,$$ then $F(x)\in\mathbb{R}$ and $F$ is ...
0
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1answer
23 views

One-point sets are G$_\delta$ in first-countable $T_1$ spaces

Can someone please verify my proof or offer suggestions for improvement? I am aware that there is a similar question elsewhere. I only need help with my proof in particular. Show that in a ...
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0answers
13 views

Mother Wavelets are symmetrical or not

Are mother wavelets symmetrical i.e can we write $\psi(-x)= \psi(x) ?$ As we know that $\psi^{a,b}(x)=|a|^{-1/2}\psi(\frac{x-b}{a})$. Then $\psi^{a,b}(0)=|a|^{-1/2}\psi(\frac{-b}{a})$. So can we write ...
0
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1answer
34 views

Infinite topological space with cofinite topology is not Hausdorff

I found a proof to the question, but mine is completely different (sort of). Is this correct? If $X$ were Hausdorff, then consider $u,v \in X$ with disjoint neighbourhoods $U, V$ that separates ...
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1answer
49 views

Exercise on characterization of free abelian groups

I was wondering if someone can please check my work on a homework problem. This is from the graduate Hungerford text. Chapter 2.1, number 3. Let $X=\{a_i\ |\ i\in I\}$ be a set. Then the free abelian ...
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2answers
29 views

$Ker(T) \subseteq V$ Is A Subspace

Let $V,W$ be a vector space over a field $\mathbb F$, and $T$ a linear transformation $T:V \rightarrow W$ $Ker(T) \subseteq V $ to prove that $Ker(T)$ is a subspace can we say that: by definition ...
3
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0answers
162 views

Proof on why $0-1+2-3+4-\ldots\neq-1/4$

When reviewing my notes on series' convergence, I thought of applying a workaround on why $\sum_{n=0}^{\infty}(-1)^nn$ should or shouldn't be $-1/4$ (I recalled this page). I started by considering ...
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2answers
25 views

Linear Transformation $T_{A}$ Is invertible $\iff$ A Is invertible

Let $T_{A}$ be the linear mapping corresponding to the matrix A, and $A \in F^{n*n}$ $T_{A}$ Is invertible $\iff$ there is $T_{A}^{-1}$ so $T_{A} \circ T_{A}^{-1}=I $ $T_{A} \circ T_{A}^{-1}(v)=v$ ...
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2answers
36 views

Proving that only the linear codes pass parity check

An exercise in my book goes as follows: Let $C$ be a binary $(n,k)$ linear code with parity-check matrix $H$. We know $Hc=0$ for all $c\in C$. Show that $Hw=0$ implies $w\in C$. My idea: Let ...
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1answer
47 views

General formula for $\frac{a^{2n+1}-a^{2n-1}}{a^n-a^{n-1}}$. Is such proof correct?

I have a very simple case: Find general formula for $\frac{a^{2n+1}-a^{2n-1}}{a^n-a^{n-1}}$. Of course dividing one by another was quite simple with outcome: $a^n(a+1)$. However I would like to prove ...
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2answers
29 views

Convergence of integral, that is absolutely convergent, proof

Can you think of any proof on convergence of improper integral, that is absolutely convergent? It is so obvious, that I really don't know where to start. Triangle inequality gives us ...
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2answers
38 views

The maximum no. of edges in a DISCONNECTED simple graph…

... on n vertices when it is not connected being equal to (1/2)(n - 1)(n - 2)... I can see that for n = 1 & n = 2 that the graphs have no edges... however I don't understand how to derive this ...
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2answers
44 views

Linear Code (9,5): Is my Parity Check correct?

I have an exercise about designing parity checks for the Hamming (9,5) group code with minimum distance $3$. I use the following notation for the generator matrix: $$ ...
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1answer
54 views

Check my proof by contradiction…

The question is to prove the following by contradiction. There does not exist a smallest positive non-zero rational number. What I tried... There does exist a smallest positive non-zero rational ...
2
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0answers
36 views

Characterisation of absolutely continuous measure on the real line

Let $\lambda, \nu$ be two Radon measures on $\Bbb R$ such that $\lambda(\Bbb R)< \infty$. Show that the following are equivalent: $\lambda \ll \nu$; $\forall \epsilon>0$ there exists ...
3
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3answers
91 views

How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for ...
2
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3answers
50 views

Show that $f(x)=x/\sqrt{x^2+1}$ is a bijection of $\mathbb R$ onto $\{ y: -1<y<1\}$

I am looking for help in regard to a practice question about functions. The question is Show that a function $f$, defined by $f(x)=x/\sqrt{x^2+1}$ , $x \in \Bbb R$ is a bijection of $\Bbb R$ onto ...