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

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1
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0answers
19 views

If $f(x)\geq 0$ for all $x \in [a,b]$ and $\alpha \in BV([a,b])$ is increasing , then $\int_a^bf d\alpha \geq 0.$

This is a proof verification question. Here, $\, f$ is continuous and $\alpha$ is of bounded variation. My only tools are the sums, for a given partition $P = \{a=x_0 < \ldots < x_n = b \}$ of $...
-4
votes
0answers
67 views

(Collatz Conjecture) Is this a valid proof? [on hold]

This was uploaded to 4chan.org/sci/ by an anonymous user this morning. In the interest of honesty, nobody should claim the prize money if this is a valid proof, as the author appears to not want it. ...
3
votes
1answer
43 views

Find all solutions of $e^{e^z}=1$ in the complex space.

Find all solutions of $e^{e^z}=1$ in the complex space. Attempt: $e^{e^z}=1$. Assuming $e^z$ is a complex number, I will start off solving $e^z=e^{x+yi}=1$: $e^x(\cos y+i\sin y)=1\Rightarrow \sin y=...
1
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1answer
25 views

Is my proof of the additivity property of Riemann integral correct?

Background I am trying to prove the following theorem. Let $f:[a,b]\to\mathbb{R}$ be a bounded function. If $c\in(a,b)$ then show that $f:[a,b]\to\mathbb{R}$ is Riemann Integrable on $[a,b]$ if $...
0
votes
0answers
7 views

$ι:U→V$ is an embedding, $Q:=ιι^*$, $L∈𝓛(ℝ^d)$, $Φ∈\text{HS}(U,ℝ^d)$ $⇒$ $\text{tr}LΦ\sqrt Q(Φ\sqrt Q)^*$ doesn't depend on $ι$

Let$^1$ $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota\in\operatorname{HS}(U,V)$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $u:\mathbb R^d\to\mathbb R$ be twice Fréchet ...
0
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0answers
18 views

Verification: Closed Set Expands to Fill Space, but Contains No Open Ball $B_\epsilon(0) $?

I have the proof that $C$ closed, convex, symmetric in Banach space $X$ and $\cup_{n \in N \setminus 0} n.C= X $ then $B_\epsilon(0) \subset C$ for some $\epsilon > 0$. I also have the proof for $...
0
votes
1answer
33 views

Increasing sequence and subsequence

I want to prove this statement, If $x_n$ is an increasing sequence and if some subsequence of it converges, then $x_n$ also converges. My proof is that suppose $x_{n(k)}$ is the subsequence that ...
3
votes
1answer
43 views

Proving that $\lim_{x\to 2}\frac{x^2-5x}{x^2+2}=-1$ using the $\epsilon$-$\delta$ definition of a limit

My attempt: $$ \left|\frac{x^2-5x}{x^2+2}+1\right|<\left|\frac{x^2-5x}{x^2+2}\right|< \left|\frac{x^2-5x}{x^2}\right|<\frac{1}{x^2}|x^2-5x|,$$ using the restriction $|x-2|<2$, so $0<x&...
6
votes
2answers
63 views

Proving $\lim_{x\to1}(x^3+5x^2-2)=4$ using the $\epsilon$-$\delta$ definition of a limit

I want to prove that the limit of $f(x)=x^3+5x^2-2$ when $x\to 1$ is $4$. So, I want to show that for any $\epsilon >0$ $\exists \delta_{\epsilon}$ such that for all $x$ that satisfies $|x-1|<\...
0
votes
1answer
28 views

Is it possible for two triangles to be different if the sides of one is equal to another?

I was reading Euclid's Elements E-book I found online and got stuck on this concept. I will just copy what I found to be very absurd. There could still be another different triangle with the same ...
-1
votes
3answers
24 views

Does this series converge or diverge? application of root test

Suppose $\sum a_n $ is convergent and $a_n > 0$ for all $n$, does it follow that $\sum \left( \frac{ 1 + \sin (a_n) }{2} \right)^n $ is convergent?? yes Since $\sum a_n$ is convergent, then $\sum ...
0
votes
4answers
35 views

Show that in the factor group $\Bbb Q / \Bbb Z$, there is an element for every $n \in \Bbb N_+$ such that the order of that element is $n$

Task: Show that in the factor group $\Bbb Q / \Bbb Z$, there is an element for every $n \in \Bbb N_+$ such that the order of that element is $n$. Solution: We take a look at the residue ...
0
votes
1answer
24 views

Applying topological definition of continuity to $f(x) = \frac{1}{x}$

I am trying to show that the function $f:\mathbb{R} \rightarrow \mathbb{R}$ defined by $$ f(x) = \left\{ \begin{array}{l} \frac{1}{x}, \, x > 0 \\ 0, \, x \leq 0 \end{array} \right. $$ is not ...
2
votes
2answers
156 views

Prove If a,b, c in N, then lcm(ca, cb) = c lcm(a,b).

Prove: If $a,b,c$ in $\mathbb N$, then $lcm(ca, cb) = c \cdot lcm(a,b)$. Assume $a$,$b$,$c \in \mathbb N$. Let $m = lcm(ca,cb)$ and $n = c\cdot lcm$. Showing $n = m$. Since $lcm(a,b)$ is a multiple ...
1
vote
0answers
31 views

Show that if $E$ is Jordan measurable then $m(A-B) \leq \epsilon$

I want to show that if $E$ is Jordan measurable then $m(A-B) \leq \epsilon$ where $A \subset E \subset B$. I think I have the right ideas but feel I am missing some details. I'd like some feedback ...
2
votes
3answers
54 views

$A \Delta C = B \Delta C$, then prove that $A = B$ where $\Delta$ is a symmetric difference operation.

I suppose that if I can prove that every element that belongs to set $A$ also belongs to set $B$ and vice versa and also any element that doesn't belong to set $A$ doesn't belong to set $B$ either and ...
3
votes
1answer
33 views

Proof of XOR properties

I want to prove the following two properties of the Nim-sum/XOR operator $\oplus$ to better understand Nim games. For the position $n = a_1 \oplus a_2 \oplus a_3 \oplus \cdots \oplus a_k = 0$, ...
3
votes
1answer
58 views

Continuous functions in the product topology on $\Bbb{R}^{\Bbb{N}}$

I'm trying to prove the following statement: Let $(X, T )$ be a topological space, and let $f : X \rightarrow \Bbb{R^{\Bbb{N}}}$ be a function, where $\Bbb{R^{\Bbb{N}}}$ has the product topology. Let ...
0
votes
1answer
45 views

Is this proof correct? Lagrange multipliers

Suppose that $f,g : \mathbb{R}^n \to \mathbb{R}$ are $C^1$ functions and $c$ is a regular value of $g$. If $a \in g^{-1}(c)$ is a minimum for $f$ restricted to $g^{-1}(c)$ then there is $\lambda \in ...
1
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0answers
21 views

Prove: $\omega(f,P)\leq \omega(f,Q)+2nL*\lambda(P)$

Let $P=\{x_{1},...,x_{k}\}$ and $P\subseteq Q$ a refinement of the partition $P$ which is due to adding one point $Q=P\cup\{y\}$, In this case both partition are the same except of on interval $[x_{i-...
2
votes
1answer
32 views

Doubt regarding a limit which is related to MVT

Let the function $f(x)$ be differentiable and $f'(x)$ be continuous in $\left(-\infty,\infty \right)$ with $f'(2)=14$ then evaluate the limit $$\lim_{x\to 0}\frac{f(2+\sin x)-f(2+x\cos x)}{x-\sin x}$...
0
votes
0answers
44 views

Can we prove invariance of dimension directly from the Jordan-Brouwer separation theorem?

Is the following proof correct? Consider spaces $\mathbb{R}^n$ and $\mathbb{R}^m$, where $n<m$, and sphere $S^{n-1}\subset \mathbb{R}^n$. Suppose that we have a homeomorphism $f:\mathbb{R}^m \...
0
votes
1answer
19 views

Proof Verification: C closed, convex, symmetric in Banach space X and $\cup_{n \in N \setminus 0} n.C= X$ then $B_\epsilon(0) \in C $.

I have an outline of the proof of this which I've expanded (correctly or otherwise) below, I'd appreciate feedback on it. (I think that C has to be closed in order to assert that $\cup_{n \in N \...
5
votes
0answers
67 views

If $|G|=p^3q^2$ then $\Phi(G)$ is cyclic for primes $p\neq q$.

I have conjectured this result for the Frattini subgroup by doing some calculations in GAP. I think this is even true if $|G|=p_1^{i_1}\cdots p_n^{i_n}$ for $i_j\leq 3$ holds, but I would like to ...
3
votes
2answers
36 views

Minimal perimeter of a triangle

Imagine a triangle with a base $[0, s]$ and a height $h$. ($s, h \gt 0$) For what orthocentre $x$ does the triangle have a minimal perimeter and how long is it? Now, the proof starts with: ...
1
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3answers
66 views

Is my proof True ? ( about Group theory, direct product )

I have a problem. It states that: Let $G$ is a group and $|G|=mn$, $(m,n)=1$. Assume that $G$ has exactly just one subgroup $M$ with order $m$ and one subgroup $N$ with order $n$. Prove: $G$ is ...
1
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0answers
32 views

Alternate Proof of Unique Lifting Property of Covering Spaces

I proved one of Hatcher's propositions on my own and my proof is quite a bit different than his. The Unique Lifting Property says: Given a covering space $p:\tilde{X} \rightarrow X$ and a map $f: ...
0
votes
1answer
21 views

Is collection of all functions I-convergent to a point form a ring?

$S$ be a set. $I$ is an ideal of $S.$ $X$ is a topological space. A function $$f: S\rightarrow X$$ is said to be $I$-convergent to a point $x\in X$ if $$f^{-1}(U)=\{ s\in S; f(s)\in U\}\in \mathscr F(...
3
votes
1answer
74 views

Is there a formal way to show that $X \cap Y \subseteq X$.

The question is in the title. It is trivial that $X \cap Y \subseteq X$. Because $X \cap Y$ only contains elements that are both in $X$ and in $Y$. So every element in $X \cap Y$ is also an element of ...
0
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0answers
25 views

$\Bbb{R}^{\Bbb{N}}_{\square}$ is not ccc

Consider $\Bbb{Z}^{\Bbb{N}} \subseteq \Bbb{R}^{\Bbb{N}}$. The set $\Bbb{Z}^{\Bbb{N}}$ is uncountably infinite, since $|\Bbb{Z}^{\Bbb{N}}|$ = $|\Bbb{Z}|^{\Bbb{|N|}}$ = $\aleph_0^{\aleph_0}$ > $2^{\...
0
votes
0answers
31 views

Tietze Extension Theorem - How does the induction work?

I am reading a version of the Tietze Extension Theorem here: https://proofwiki.org/wiki/Tietze_Extension_Theorem There was a Lemma that says: And then it was repeatedly applied: How was the ...
0
votes
1answer
53 views

Are functors (from small categories) functions?

I am looking for either (1) confirmation that the following is true, (2) the mistake making it false pointed out to me: Let $F:\mathcal{C} \to \mathcal{D}$ be a functor from a small category $\...
1
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0answers
41 views

Is this metric space normable?

Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by $$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$ ...
2
votes
3answers
36 views

$(R,\mathcal m)$ be a Noetherian local ring and let $P$ be a prime ideal of $R$. If $P^2$ is a prime ideal of $R$, then $P=0$

Let $(R,\mathcal m)$ be a Noetherian local ring and let $P$ be a prime ideal of $R$. If $P^2$ is a prime ideal of $R$, then $P=0$. I was thinking to use Nakayama lemma as: $R_P$ is local with $PR_P$ ...
0
votes
3answers
46 views

Proving if $F^{-1} $ is function $\Rightarrow F^{-1}$ is $1-1$?

Let F be a function from set A to set B. If $F^{-1}$ is a function, then $F^{-1}$ is one to one. Prove: If $F: A \rightarrow B $ and $F^{-1}$ is a function, then F is one-to-one. Proof: ...
2
votes
1answer
29 views

Integral of Simple Functions converges to Integral of Measurable Function

Let $f$ be a measurable function and $E_{n,m} = \{x : \frac{m}{2^n} \leq f(x) < \frac{m+1}{2^n} \}$. Prove: $$\lim_{n \to \infty} \sum_{m=1}^{\infty} \frac{m}{2^n} \mu(E_{n,m}) \to \int f \, d\mu$...
2
votes
2answers
32 views

The Greatest Number of Edges on a Bipartite Graph

Let $G$ be a bipartite graph on $p$ vertices. Find a formula in terms of $p$ that determines the greatest number of edges that $G$ could have. Prove that this formula is correct. Let $V$ be the set ...
0
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0answers
14 views

On $\sum_{\substack{\zeta(\frac{1}{2}+i\gamma)=0\\0<\gamma<T}}\prod_{n=1}^\infty \left| 1-\frac{(\gamma\log x)^2}{n^2\pi^2}\right|$ as $O(\log x)$

On assumption that the identity (2) for a representation of $\pi(x)$ holds, see here Two Representations of the Prime Counting Function in this site Mathematics Stack Exchange, and since using the ...
0
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0answers
15 views

Proof regarding little o for vector field

I'm struggling with one part of a proof of a theorem. Let $\gamma(t) : A \subseteq \mathbb{R} \rightarrow \mathbb{R}^m$, with $\gamma \in \mathscr{C}^1(A)$ (hence $\gamma$ differentiable). ...
1
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0answers
20 views

Proof of equivalence between limit of a vector field and limit of a scalar field

I have a doubt with a proof regarding the following implication. Consider $F=(f_1,..,f_m): A \subset \mathbb{R}^n \rightarrow > \mathbb{R}^m$ and $\bar{x}$ a limit point for $A$, then $$\...
2
votes
1answer
59 views

Check whether my proof is correct or not.

The problem is : If the series $\sum a_n ^2$ and $\sum b_n ^2$ be both convergent, prove that the series $\sum a_n b_n$ is absolutely convergent. Using A.M. > G.M. we have $(a_n ^2 + b_n ^2)/2 \...
1
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2answers
29 views

Alternative proof: show that any metrizable space $X$ is normal - Part 2

This is a follow up to one of my earlier questions I am reading some stuff online and saw a proof as follows Per a comment in Part 1 in linked, We know that $d(C_1,C_2)$ could easily be zero ...
1
vote
1answer
35 views

Proving that a prime ideal $p \subset R$ yields a prime ideal $p[x] \subset R[x]$

I'm curious as to whether I can have my proof critiqued. Proposition : Let $\mathfrak{p}$ be a prime ideal in a ring $R$. Show that $\mathfrak{p}[x]$ is a prime ideal in $R[x]$. Proof : Suppose $\...
3
votes
3answers
100 views

“Alternatives” to Natural Transformations

I would like someone to either (1) point out the mistake in what follows or (2) confirm what is said is correct. This would be accomplished by addressing the part in yellow only. The rest of the ...
1
vote
0answers
22 views

Proof verification: Show that the fixed field is $\mathbb{Q}(\sqrt{3})$

Let $H$ be the subgroup $\{i,\alpha\}$ of $\text{Gal}_{\mathbb{Q}}\mathbb{Q}(\sqrt{3},\sqrt{5}),$ where $i$ is the identity map and $\alpha$ is defined as $\alpha(\sqrt{3})=\sqrt{3}$,$\alpha(\sqrt{5})=...
6
votes
6answers
114 views

Proving that ${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $

How can I prove that $${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $$ I tried the following: We use the falling factorial power: $$y^{\underline k}=\underbrace{y(y-1)(...
2
votes
5answers
89 views

Monotonicity of the sequence $(a_n)$, where $a_n=\left ( 1+\frac{1}{n} \right )^n$

Define $a_n=\left ( 1+\frac{1}{n} \right )^n$ for $n\geq 1$. I want to show that it is increasing. First, we have $$\frac{a_{n+1}}{a_n}=\left ( \frac{1+\frac{1}{n+1}}{1+\frac{1}{n}} \right )^n\left ( ...
3
votes
1answer
46 views

$\text{ Proving }\; A \subseteq \Bbb R \text{ A is bounded above} \Rightarrow A^c \text{ is not?} $

Prove: Let $A \subseteq \Bbb R$. Prove that if $A$ is bounded above, then $A^c$, the complement of $A$ is not bounded above. $ A^c = $ those element of the universe that are not in A. $ \Bbb R =$ ...
0
votes
1answer
56 views

Can I show that $\int_{\gamma(0;r)} \frac{1}{z-a} dz = 0$ when $|a|>r>0$ without using Cauchy Theorem?

I encountered this problem as a previous result of an exercise in a text book way before proving Cauchy Theorem, so I think there must be another way to prove it without it. Show that $\int_{\...
3
votes
0answers
40 views

Derivative of Exponential map on manifolds

I'm trying to compute the derivative of the map $f:\Sigma\times [0,\delta)\to M$ given by $$f(p,t)=\exp_p tN(p),$$ in $X\in T_p\Sigma$, where $(M^n,g)$ is a Riemannian manifold, $\Sigma\subset M$ a ...