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

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Analysis Proof- different conditions.

A continuous function on $[a,b]$ is also uniformly continuous on $[a,b]$. The following tries to illustrate what happens when the interval is not closed: Show: $f(x) = \frac{1}{x} $ is not ...
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1answer
48 views

Is this alternative proof of Theorem 3.7 (“Baby” Rudin, Ch. 3) correct and, if so, well written?

Rudin, in his Principles of Mathematical Analysis, proves the following theorem: The subsequential limits of a sequence $\{p_n\}$ in a metric space $X$ form a closed subset of $X$. I've tried to ...
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0answers
27 views

Limit Points and Convergence of Sequences.

Let $E \subseteq \mathbb{R}$ (or $\mathbb{C}$). A point $p \in \mathbb{R}$ (or $\mathbb{R}$ ) is called a limit point of $E$, if $\forall \epsilon > 0$, $\exists z \in E$ such that $0 < |z − p| ...
7
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1answer
58 views

$a \in G$ commutes with all its conjugates iff $a$ belongs to an abelian normal subgroup of $G$ [duplicate]

Let $a\in G$, where $G$ is a group. Prove that $a$ commutes with each of its conjugates in $G$ if and only if $a$ belongs to an abelian normal subgroup of $G$. This is what I did: $"⟹"$ First, ...
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1answer
34 views

What is wrong with this inductive proof?

I have found a startling proof by induction which is clearly wrong. Let L(n) represent Lucas numbers. L(0)=2, L(1)=1 L(n) = L(n-1) + L(n-2) Let F(n) denote a Fibonacci number. F(0) = 0, F(1) = 1, ...
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1answer
69 views

The proof of theorem 3.19 from baby Rudin

If $s_n\leqslant t_n$ for $n\geqslant N$, where $N$ is fixed, then $$\liminf_{n\to \infty} s_n\leqslant \liminf_{n\to \infty} t_n$$ $$\limsup_{n\to \infty} s_n\leqslant \limsup_{n\to \infty} t_n$$ ...
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5answers
49 views

Help with proof that $\sum_{n \in \Bbb{N}} \frac{1}{an + b}$ also diverges?

We know that $\sum_{n \in \Bbb{N}} \frac{1}{n}$ diverges. So it seems likely that $\sum_{n \in \Bbb{N}} \frac{1}{a n + b}$ will for any real $a, b$. I'm having trouble proving it just for the ...
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2answers
31 views

Analysis Limit- Function Proof

A) For every sequence ($p_n$) in $E$ such that $p_n \not= p$ and $p_n \rightarrow p$ as $n \rightarrow \infty$ we have that $f(p_n) \rightarrow l$ as $n \rightarrow \infty$ . ($E \subseteq ...
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1answer
33 views

Straight line through $(a,b)$ with slope $m$ is the graph of the function $f(x) = m(x-a) + b$

Spivak's Calculus Chapter 3 Problem 6 says: Show that the straight line through $(a,b)$ with slope $m$ is the graph of the function $f(x) = m(x-a) + b$. Since the slope in a graph of a line is ...
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0answers
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Complex Sequence Convergence

Claim : If complex sequence $z_n$ converges then $|z_n|$ converges Proof: Let $z_n =x_n +i y_n$ where $x_n$ and $ y_n$ are real sequences. If $z_n $ converges to $(L_1+i L_2)$ $ \forall \epsilon ...
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2answers
47 views

A plane contains a set of marked points, such that any three can be covered by a unit disk. Prove that the entire set can be covered by a unit disk.

A set of points is marked on the plane, with the property that any three marked points can be covered with a disk of radius 1. Prove that the set of all marked points can be covered with a disk ...
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1answer
26 views

Show that $\left \{ \bigcup_{i\in I}A_{i}:I\subseteq \{1,\dots, n\} \right \}$ is a $\sigma$-algebra

Let $\{A_{i}\}_{i = 1}^{n}$ be a family of pairwise disjoint subsets of $X$. It is said that $$\mathcal{F}:=\left \{ \bigcup_{i\in I}A_{i}:I\subseteq \{1,\dots, n\} \right \}$$ is a $\sigma$-algebra. ...
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1answer
52 views

Do proof assistants like Coq really need to actually perform computations to prove n <= m, or is there a more optimal algorithm?

For example, trying to prove that 100,000 <= 1,000,000. But Coq has a stack overflow, meaning it's actually trying to perform the 100k computations. ...
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1answer
27 views

Order of a corrector-predictor method

Given an explicit method: $$ x_{i+1} = x_i+ h \Phi(t_i,x_i,h) $$ as predictor method and an implicit method: $$ x_{i+1} = x_i + h \Psi(t_i,x_i,x_{i+1},h) $$ as corrector method, it follows that $$ ...
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2answers
41 views

Calculus Spivak. Chapter 1. Question 1. (i) or are there many ways of skinning a cat

I'm taking on Spivak's Calculus a little later on in life via self-study as i'm looking to improve my CS abilities and have always been interested in Maths but unfortunately didn't have the chance ...
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1answer
22 views

If $S\leq G$, prove that $S\unlhd G \iff \gamma (S) \leq S$ for every conjugation $\gamma$

If $S\leq G$, prove that $S\unlhd G \iff \gamma (S) \leq S$ for every conjugation $\gamma$ I have proven the forward direction but I am not sure that the way I prove the converse is true. Let $g\in ...
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4answers
63 views

Prove that if $A$, $B$ are countable, then $A \times B$ is countable?

Is $A\times B$ referring to the axis here? So an $X$ and $Y$ coordinate plane? $A$ is countable, therefore a bijection occurs from $A \rightarrow \mathbb{N}$. $B$ is countable, therefore a bijection ...
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2answers
29 views

Prove or disprove the following asymptotic relations

$P(x) = 2^x$ Prove or disprove that $p(n^3 + 4) \in O\left(p\left(n^3\right)\right)$ $2^{(n^3 + 4)} \in O(2^{n^3})$ $\lim_{n \rightarrow \infty} \space \frac{2^{n^3 + 4}}{2^{n^3}}$ using ...
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1answer
49 views

Let $(s_n)$ be a sequence in $\mathbb{R}$. Prove $\lim_{n \to \infty}s_n=0$ if and only if $\lim_{n \to \infty}|s_n|=0$

First, assume that $\lim s_n=0.$ This implies that for any given $\epsilon > 0$, $\exists$ an $N$ such that for $n>N,|s_n-0|< \epsilon$. $|s_n-0|=|s_n|<\epsilon$ and $|s_n|=||s_n|-0| ...
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0answers
57 views

If $A$ and $B$ are arbitrary $m\times n$ matrices, show that $^t(A+B)= {}^tA+{}^tB$?

I'm reading Lang's: Introduction to Linear Algebra. There is this exercise: If $A$ and $B$ are arbitrary $m\times n$ matrices, show that $^t(A+B)= {}^tA+{}^tB$ I did the following: ...
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1answer
39 views

Is this a correct proof that $\tau(n)$ is eventually smaller than any $n^{\delta}$, $\delta>0$?

Is this a correct proof that $\tau(n)$ is eventually smaller than any $n^{\delta}$, $\delta>0$? I'm not even sure about the statement, let alone the proof. Let's first proof this result: $\tau(n) ...
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2answers
79 views

prove $\langle f(x),f'(x)\rangle = 0$

Let $f: R \to R^n$ be a differentiable function such that $\forall x \quad||f(x)|| = 1$ prove that $\forall x \quad \langle f(x),f'(x)\rangle = 0$ i thought of the following proof but not sure it ...
0
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1answer
46 views

Which of the following sets are countable?

a) $[0,1] \cap \mathbb Q$ b) $P(\mathbb Q)$ c) $\mathbb R \setminus \mathbb Q $ d) $\{(a, b) ∈ \mathbb R\times\mathbb R | a, b \in\mathbb N\}$ I answered a) and d) a) any intersection between ...
0
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5answers
49 views

Is $F:R\to[0, ∞)$ where $F(x) = e^x$ a bijection?

Is $F:R\to[0, ∞)$ where $F(x) = e^x$ a bijection? For a function to be surjective, the function must hit all elements belonging to the CODOMAIN (Which is $[0,∞)$ right?) or does it simply have to ...
0
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0answers
20 views

Partial sum of a product over an arbitrary sequence.

Below is an equation a friend showed me, but was unable to prove. After struggling with it for a bit I was unable to as well. After failing to show this for, say, N=2 Im pretty sure the equation is ...
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1answer
41 views

Which of the following are bijections?

• $f : \mathbb{Z} → \mathbb{Z} \\ f(x) = x^5 - 3$ • $g : \mathbb{R} → \mathbb{R} \\ g(x) = x^5 - 3$ • $h : \mathbb{Q} → \mathbb{Q} \\ h(x) = x^5 - 3$ • $F : \mathbb{R} → [0, ∞) \\ F(x) = e^x$ ...
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3answers
24 views

Let $A = \mathbb{Z}$, $B = [−1, \pi]$, $C = (2, 7)$. List all elements of $A \cap (B^c \cap C)$.

After working it out on a number line, I got: $\{4, 5, 6\}$. As it stands, the expression contains the integers that do not belong to the set $B$ that cross into $C$. This would result in $4, 5, 6$. ...
1
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1answer
26 views

Define the image of the function $f :\Bbb Z \times \Bbb N →\Bbb R$ given by $f(a, b) = \frac{a−4}{7b}$?

$\Bbb Z$ - integers $\Bbb N$ - natural numbers (starting from 1) $\Bbb R$ - real numbers I believe the answer is the set of real numbers ($\Bbb R$), seeing as $b$ will not equal $0$ as the set of ...
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2answers
39 views

Verifying a Proof for Spivak's Calculus Question (Chapter 2 Problem 9)

It says "Prove that if a set $A$ of natural numbers contains $n_0$ and contains $k+1$ whenever it contains $k$, then A contains all natural numbers $\ge n_0$". Am I allowed to construct another set ...
3
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1answer
44 views

Convergence of the integral $\int_0^1 \frac {1}{x\sqrt {1+x^\beta}}dx$

Is my integral-convergence contradiction proof valid? I have to brush up on my proof making. I am a little rusty. I was not sure if the following really held up. I wanted to prove the following is ...
2
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2answers
73 views

prove if f(x) has an infinite limit then limit of 1/f(x) is = 0

I wanted to ask if someone can do me the favor pointing out the mistakes I might of made in proving the theorem below. Also is there a way to prove the theorem without using the definition of limits? ...
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2answers
65 views

Proving $\frac{1}{4n}$ converges to $0$ in epsilon-delta form.

$$(\forall\varepsilon>0)(\exists K)(\forall x)(x>K\Rightarrow\frac{1}{4x}<\varepsilon)$$ Here is my attempted proof: Let $\varepsilon>0$ be a real number. Let $K$ be ...
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1answer
34 views

Divisibility proofs for greatest common divisor

I am studying divisibility and greatest common divisors. I have reached a section where I need to prove properties. My question is: are my proofs substantial? Or do I need to add to them? Below are ...
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1answer
180 views

Prove $\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$ for-

Let $n$ be a positve integer. Prove that$$\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$$ for each non-negative ...
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2answers
23 views

Confusion with the reconstruction conjecture?

After reading about the reconstruction conjecture for graphs, I came up with what I thought was a proof by contradiction. Consider the class $T$ of (isomorphism classes of) finite graphs, and the ...
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1answer
61 views

Mean value theorem for vector valued function (not integral form)

Let $f:U\to\mathbb R^m$ be differentiable with $U\subseteq \mathbb R^n$ being open and convex. If $f$ is absolutely continuous, then by fundamental theorem of calculus we have following version of ...
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0answers
33 views

Proof Verification: Show $\lim_{x\rightarrow\infty}f(x)=0$ given $f\in L^1$ and $f$ absolutely continuous.

so I am working through a packet of old exams and have came across three that are very similar. I was able to produce a proof (possibly wrong?) that works for all three, but it doesn't use all their ...
3
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0answers
67 views

Find two homeomorphic topological spaces and a bijective continuous map between them which is not homeomorphism.

I'm aware that it is duplicate, but I'd like to know whether my example is appropriate or not. Let our function $f$ be on the set $\mathbb{Q}\cap\mathbb{Z}$ induced by standard topology of a line. ...
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0answers
11 views

Solution of $\sigma(p^3q^2)=2\varphi(p^3q^2)$?

Please suggest me if I made any mistake in the following: It is given that $\sigma(p^2q^2)=2\varphi(p^2q^2)$ has no solution in positive integer if $p, q$ are distinct prime. Let us see if ...
0
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1answer
37 views

Induction proof question

Show by induction that for all integers n $\ge$ 1 $$ \sum_{i=1}^n i3^i = \frac{3(2n3^n-3^n+1)}{4} $$ Starting with n = 1 will give me LHS = 3 and RHS = 3. Inserting n = p gives $$\sum\limits_{i=1}^p ...
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0answers
48 views

Debunking an elementary proof of FLT

José Cayolla: Fermat's Last Theorem admits an infinity of proving ways and two corollaries. arXiv:1507.06989 [math.GM] I don't usually devote so much time to "crackpot papers", but I have a ...
3
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4answers
100 views

Prove that $n!>n^2$ for all integers $n \geq 4$.

I am working on induction problems to prep for Real Analysis for the fall semester. I wanted proof verification and editing suggestions for part (a), and assistance understanding part (b). For part ...
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1answer
34 views

Let $n$ and $p$ be positive integers. Show that $n$ can always be expressed in the form $n=pq+r$

I would have thought this would have been on here somewhere. Here I go. Let $S$ be the set of positive integers $n$ which can be expressed in the form $n = pq + r$ where $ 0 \leq r < p.$ where ...
3
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2answers
67 views

Is the following a correct logical proof?

A → (F ∧ P) ~A → (S ∧ R) ~R ∴ P     assume ~P         assume A         F ∧ P ...
2
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1answer
34 views

$C^1(\bar \Omega)$ is a Banach space

My professor gave a proof of the completeness of $(C^1(\bar \Omega),\|\cdot \|_{C^1})$ based on the fundamental theorem of calculus. I though about an alternative and I would like to know whether this ...
1
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1answer
75 views

Proof check $(\log\log n) /(\log n) $ approaches zero

Proof : If $|a| < 1$ then $(na^n)$ is a null sequence therefore if $b>1$ then ${n\over (b)^n}$ is a null sequence.There is always an $m$ such that for every $n > m$ $${n\over (b)^n} ...
4
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2answers
62 views

Prove: the countable product of regular topological spaces is regular.

Prove: the countable product of regular topological spaces is regular. Label the countable product of $X_i$ as $X$. Given $x \in X$ and $U$ a closed set s.t. $ x \notin U$, let's find disjoint ...
2
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1answer
59 views

Is the sequence $S_n=1+\sum_{m=2}^n \prod_{k=m}^n a_k$ bounded?

I have the following sequence $$S_n=1+\sum_{m=2}^n \prod_{k=m}^n a_k$$ where $\{a_k\}$ is a real positive sequence with the property that $$\lim_{n\to \infty}\left(\prod_{k=1}^n ...
0
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1answer
17 views

Determining cardinality and inverse

Let the function $\chi: P(Z) \to P(Z)$ be defined by $\chi(B) = B^c$ for any $B \in P(Z)$. (In other words, $\chi$ sends a subset $ B \subseteq Z$ to its complement, $B^c$, i.e. the set $Z - B$.) ...
2
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1answer
74 views

A group is abelian if and only if the center of the group is all the group

Isn't it the same to say that a group is abelian, and that the center of the group is all the group? I have an exercise to prove that this is true, and it's exactly one stroke for each direction of ...