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

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2
votes
1answer
26 views

Interior of a set is the largest open subset - proof verification

I want to prove the following: Let $(X, \mathcal{T})$ be a topological space. Let $A \subset X$ be a subset. Then $\mathrm{int}A$ is the largest open subset of $A$. Our definition is ...
1
vote
1answer
63 views

Automorphism group of a graph = Automorphism group of that graph's adjacency matrix?

Is automorphism group (or set) of a graph $G$ equal to the automorphism group (or set) of adjacency matrix of $G$? Example: $G_1, G_2$ are separate graphs where $G_1^{\pi}= G_2$ and $ G= \bar ...
0
votes
0answers
18 views

Solving large set of PDEs and verfication of solution

Please comment on solution of following system of PDEs, I have checked solution several times but could not find any error but there are problems when I proceed with solution obtained as below. I ...
1
vote
1answer
16 views

Proof of discrete probability monotone convergence

I am trying to show that for a sequence of random variables defined on a sample space $\Omega$ $$0\leq X_1(\omega)\leq X_2(\omega \leq ......\leq X_{n}(\omega)...$$ for all $\omega\in\Omega$, with ...
2
votes
2answers
35 views

Please verify my delta-epsilon limit proof of $\lim_{x\to1} x^2 -6x = -5$

I want to prove that the limit exists using the delta-epsilon limit definition. Please somebody verify my solution. The given problem is $$\lim_{x\to1} x^2 -6x = -5.$$ My solution: Let $ε>0$. ...
2
votes
0answers
19 views

Expected number of cycles of length $k$ in a random graph. My simple (too simple?) solution

I attempted this on my own and got a fairly simple solution. However, after reading proofs here and here, I feel like I have massively over simplified the problem. I understand the other solutions, ...
0
votes
1answer
23 views

Graph of a continuous function $f:M\to N$ is a closed subset of $M\times N$

I need to prove that the graph of a continuous function $f:M\to N$ is a closed subset of $M\times N$. $N$ is a metric space. I think I'm supposed to use this result. So, that's what I did: $Graph(f) ...
2
votes
1answer
46 views

If $\lim_{x\to a} g(x) = M$, show that there exists a number $\delta > 0$ such that $0 < |x - a| < \delta \Rightarrow |g(x)| < 1 + |M|.$

The hint given was to take $\epsilon = 1$ for the formal definition of a limit. Doing this means that $|g(x) - M| < 1.$ Using the triangle inequality, you get $$|g(x)| = |(g(x) - M) + M| \le |g(x) ...
1
vote
0answers
25 views

How to fix this proof that isomorphic varieties have the same dimension? Is it possible?

I am trying to prove the following: Show that affine algebraic varieties that are isomorphic have the same dimension. For completeness let's state the definitions: Let $V,W$ be varieties. ...
0
votes
0answers
29 views

Proof by induction of Gronwall's inequality

I've an exercise which is the following: Gronwall’s Inequality Let $A > 0, B \geq 0$. Let $(\epsilon_j)_{j \in \mathbb{N}}$ be a sequence of real numbers with $$|\epsilon_{j+1}| ...
0
votes
0answers
34 views

Solution verification: Curve is given by points $(t,t^2, t^3)$

I tried to solve the following exerice: Show that the twisted cubic curve corresponding to the affine variety $V(x^2 - y)\cap V(x^3 - z)$ consists of all points in $\mathbb A^3$ of the form ...
1
vote
3answers
67 views

Find the Mistake to this Problem

Let $(a,b)$ be an open interval of real numbers and let $c \in (a,b)$. Describe an open interval $I$ centered at $c$ such that $I \subseteq (a,b)$. Here is the proposed solution to the problem: Let ...
1
vote
1answer
20 views

Affine set characterization

We say that a set $S\subset \mathbb{R}^n$ is $\textit{affine}$ if $$ (\forall x_1,x_2 \in S)(\forall \lambda \in \mathbb{R})\ \lambda x_1 + (1-\lambda)x_2 \in S. $$ Now, there is a theorem which ...
0
votes
3answers
18 views

Help with logical equivalences and proving tautology

I've been wracking my brain trying to figure this out, but I don't know what to do after a certain point. I'm trying to prove whether or not this is a tautology: $$ [(p\wedge r)\wedge (p\rightarrow ...
3
votes
5answers
60 views

Solution of $x^y=y^x$ and $x^2=y^3$

Solve the given set of equations: $x^y=y^x$ and $x^2=y^3$ where $x,y \in \mathbb{R}$ Would any other solution exist other that $x=y=1$ because I think $x^2=y^3$ will only be true for $x=y=1$ or ...
1
vote
1answer
36 views

Linear algebra proof with exchange theorem

Assume the Exchange Theorem and prove the following: Assume the vector space V is finitely generated. Then there is a natural number n such that the length of a linearly independent sequence is less ...
3
votes
0answers
41 views

The tangent map of multiplication - Maurer-Cartan form

Question: Consider the multiplication map $\mu : G \times G \to G$ of a Lie group. So on the tangent level we have a map $T(G \times G) \to TG$. Making the proper identification $T(G\times G) ...
2
votes
1answer
53 views

Possible mistake in Apostol's book: “An introduction to analytical number theory” (?)

On page 132 of Apostol's "An introduction to analytical number theory" : Theorem 6.6: Let $G'$ be a subgroup of a finite abelian group $G$, where $G' \neq G$. Choose an element $a \in G$, $a \notin ...
2
votes
0answers
49 views

Proof that $(a,b)\subseteq \mathbb{R}$ is an open set

To prove $(a,b)$ is an open set in $\mathbb{R}$ using the definition of an open set, we will show that for each $c \in (a,b)$, a neighborhood $N_{\epsilon}(c)$ centered at $c$ exists, such that ...
0
votes
1answer
70 views

f continuously differentiable implies f is Lipschitz on compact subsets

It is a more general form of the question here, only here $U$ is not a convex set but an open and connected subset of $\mathbb{R}^n$. I need to show that $f$ is $M$ Lipschitz on any compact $K \subset ...
1
vote
0answers
13 views

Proving three asymptotic identities (Murray (1984)'s Exercise 1.1.4)

(Context: I'm self-studying Murray (1984). I learned (and have forgotten quite a lot of) real and complex analysis. I'm willing to relearn and to look up references.) Problem: if $f=O(g)$, show that ...
1
vote
0answers
18 views

Solving or knowing something about a non-linear PDE which is “almost” linear?

Let $a>0$ be fixed. I have the following PDE: $u=u(t,x)$, $t\in [0,1]$, $x\in \mathbb{R}$, $$-\partial_t u = |\partial_x u| + \frac{1}{2}\partial_x^2 u, \quad ...
1
vote
0answers
40 views

Proof: $X$ is Hausdorff if and only if the diagonal $\Delta$ is closed in $X\times X$.

Prove that $X$ is Hausdorff if and only if the diagonal $\Delta$ is closed in $X\times X$. This exercise appeared on a previous exam in my course, and also in Munkres. Here's my attempt: First I ...
3
votes
1answer
110 views
+50

Show that $(L^{p},\|\|_{p})$ is a Banach space.

Show that $(L^{p},\|\|_{p})$ is a Banach space. My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So, ...
1
vote
1answer
18 views

How to improve my proof and whether or not one condition in the statement is important in writing the proof?

A simple graph $G$ is connected iff for every partition of the vertices into two non-empty sets $X$ and $Y$, there is a vertex $x\in X$ and a vertex $y\in Y$ such that $xy$ is an edge of $G$. My ...
-4
votes
0answers
33 views

Can someone tell me if constitutes enough proof to solve this infinite product?

I have a project do for my Calc II class where we must prove that $\lim_{n\to\infty}\prod_{k=1}^n(1-a_k)=0$ where $\{a_k\}_{k=1}^\infty$, $1>a_k>0$, $\sum_{k=1}^\infty a_k=\infty$. ...
1
vote
0answers
55 views

Is Doobs theorem of binary rank really true?

The theorem states that any adjacent matrix of the line graph of a connected graph has a binary rank n-1 if the order, n, of the graph is odd. I have pondered about this and found that it doesn't ...
1
vote
4answers
46 views

Is this a correct way to solve this high school coordinate geometry question?

Here's the question: Given point $A$: $(-3;-1)$ Given point $B$: $(3;7)$ Given point $Z$: $(x;0)$ Find the $x$ coordinate of point $Z$ so that the angle of view of AB segment is $90$ ...
1
vote
3answers
42 views

What Proof Strategy to use

I have this theorem(see below) that I am trying to prove. However, I am struggling with how to get started; I don't understand what which proof strategy to use like proof by contradiction, if P then ...
2
votes
0answers
16 views

Difference modes of convergence of a sequence of independent Bernoulli random variables

Suppose $(r_n)_{n \geq 1}$ is a sequence in $(0,1]$, $(X_n)_{n \geq 1}$ is a sequence of independent Bernoulli random variables such that: $P(X_n=0) = 1 - r_n, P(X_n = \frac{1}{r_n}) = r_n$. Show ...
0
votes
1answer
27 views

Showing $\mathbb{B}_{\mathbb{Q}}$ is a bases for $\mathbb{R}_{\text{usual}}$

Show that the collection $\mathbb{B}_{\mathbb{Q}} := \{(p, q) \subseteq \mathbb{R} : p, q \in \mathbb{Q}, p < q \}$ is a basis for the usual topology on $\mathbb{R}$. Solution: We know that ...
0
votes
2answers
29 views

$f$ is bijective, show that $h(x)=\left(f(x); g(x)\right) \rm{\ is\ bijective\ } \iff G $ is Singleton

Let E, F and G be three sets ($E\neq 0;F\neq 0,G\neq 0 ) $ Let $h$ defined by : $$\begin{align} h \ \colon\ E & \to F\times G\\ x & \mapsto h(x)=\biggl(f(x);g(x)\biggr). \end{align}$$ ...
1
vote
0answers
44 views

$\int_{-\infty}^\infty \frac{dz}{z - z_0}$ by contour integration

Consider the integral $\int_{-\infty}^\infty \frac{dz}{z - z_0}$. It has a simple pole at $z = z_0$. Assume $\Im (z_0) < 0$ so the pole is in lower half-plane. Divide $$ \oint_{C_0} = \int_{-R}^R ...
1
vote
2answers
28 views

Is $h(x)=\left(x^2; 1_{[0,\infty)}(x)\right)$ an injective function?

Let h defined by : $$\begin{align} h \ \colon\ \mathbb{R} & \to \mathbb{R}^{2} \\ x & \mapsto h(x)=\biggl(f(x);g(x)\biggr). \end{align}$$ and $$\begin{align} f \ \colon\ \mathbb{R} ...
0
votes
2answers
32 views

Proof series decreases by induction

I have a question regarding a proof by induction. We have to see whether or not the following series converges. $$U_n = \frac{1 \cdot 4 \cdot 7 \cdots (3n - 2)}{2 \cdot 5 \cdot 8 \cdots (3n-1)}$$ I ...
0
votes
1answer
32 views

In the co-finite topology and the co-countable topology, must $X$ be finite or countable?

Recall $\tau_{co-finite} = \{U \subseteq X| X\backslash U \text{ is finite}\}\cup\{\varnothing\}$ $\tau_{co-countable} = \{U \subseteq X| X\backslash U \text{ is countable}\}\cup\{\varnothing\}$ ...
3
votes
2answers
37 views

Could anybody please check my proof about connected graph?

I have written a proof of the following statement but not sure whether it is correct or not. Let $G$ be a connected graph and each vertex has even degree. Show that if we remove ANY edge of the graph ...
1
vote
2answers
33 views

Dirac-Delta function representation - infinite sum involving trigonometric identities

Proof of the identity: $$\delta (x-x') = \sum_{n=0}^{\infty} \Big\{ \cos[n \pi(x-x')] - \cos[n \pi(x+x')] \Big\} \tag{1}$$ I can intuitively tell that this function is $\infty$ for $x=x'$, and ...
0
votes
0answers
39 views

Topology bases for $\mathbb{R}_{\text{usual}}$

I'm trying to compile correctly formulated solutions to common topology questions as a summer project. I'm not very confident in my proof writing abilities so I'm going to post my solutions here for ...
0
votes
1answer
13 views

Is the weak-* topology on a topological vector space Hausdorff?

Let $V$ be a topological vector space and $V^*$ be the space of linear functionals induced with the weak-* topology. Can we say that $V^*$ is Hausdorff? Here is my attempt: Let $\lambda\ne\lambda'\in ...
1
vote
2answers
40 views

Constructing topology on $\Bbb{Z}$

Fix an infinite subset $A$ of $\mathbb Z$ whose complement $\mathbb{Z}\setminus A$ is also infinite. Construct a topology on $\mathbb{Z}$ in which: (a) $A$ is open (b) Singletons are never open (i.e ...
4
votes
1answer
40 views

Show that if $(\sum x_n)$ converges absolutely and $(y_n)$ is bounded then $(\sum x_n y_n)$ converges

This is the exercise 2.7.6 of the book Understanding analysis of Abbott, I want a check of my proof and if is needed additional information to complete it. a) Show that if the sequence $(\sum ...
0
votes
2answers
24 views

Check my proof on showing a graph with each vertex's degree at least $e$ has every tree with $e$ edges a subgraph

Let $T$ be a tree with $e$ edges and $G$ be a simple graph such that ech vertex has degree at least $e$. We need to show that $T$ is a subgraph of $G$. I tried to prove this by induction. The base ...
1
vote
1answer
25 views

If $X$ is totally bounded then every sequence contains a Cauchy subsequence

I attempted the proof, I just want to see if it is correct: Suppose $X$ is totally bounded and $(x_n)$ is a sequence in $X$. Then $(x_n)$ has a subsequence contained in a ball of radius $1/2$. This ...
1
vote
0answers
19 views

Regular values of $g(x,y)= x^2 - y^2$

I am doing some very introductory studying about manifolds. I wanted to check I was getting the right end of the stick through this example. Could anyone verify/correct my solution to the following ...
1
vote
0answers
33 views

Details on Proving that $\lim_{n \rightarrow \infty}\int_{-M}^M f(x) \cos (nx) dx=0$ Using Density of Step Functions

I was working on a question very similar to this post: Show that $\int_{-\pi}^\pi ~f(x) \cos (nx) \mathrm{d}\mu(x)$ converges to $0$ . I want to show that $\lim_{n \rightarrow \infty}\int_{-M}^M ...
0
votes
1answer
32 views

Is the conjunction of all necessary statements sufficient? What about the converse?

A necessary condition for consequent $q$ is a proposition $p$ such that: $$\neg p \implies \neg q$$ let $P:= \{p_i: \neg p_i\implies \neg q\}$ What I want to know is if $$\bigwedge_{p_i\in P} p_i ...
1
vote
1answer
41 views

Prove $n < 2^n$ for all $n \geq 0$ using induction.

Please verify for me? Base case: $n = 0 $ $0 < 2^0$ $0 < 1$. This is true. Inductive step: Suppose $n \geq 0$. Assume $P(k)$ is true if $k = n$. We must deduce that $P$ holds for $k+1$. $n ...
2
votes
5answers
58 views

Proving that the groups $S_3$ and $D_6$ are isomorphic [duplicate]

In particular, $S_3$ is the group of permutations of $\{1,2,3\}$, and $D_6$ is the dihedral group of symmetries of the triangle (written as $D_{2\cdot 3}$). In generator-relation form, $D_6 = ...
1
vote
2answers
44 views

Does the Hausdorff property hold on closed subsets of $\mathbb{R}^n?$

I am trying to prove that given disjoint closed $A,B\subseteq \mathbb{R}^n$, there exist disjoint open $U,V$ containing $A,B$ respectively. In other words that we can take the Hausdorff property to ...