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

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2
votes
4answers
66 views

Seating four girls and two boys in a row such that the boys do not sit together

If $2$ boys are never to sit together and $4$ girls and $2$ boys are to sit in linear line.? Then total number of such arrangements is: My solution: The total number of linear arrangements is $6!$ ...
0
votes
0answers
29 views

Weakly compact closed balls in reflexive space

If it is given that $X$ is a reflexive Banach space. Let $K \subset X$ be a norm closed and norm bounded convex set. I want to show that $K$ is weakly compact. I have the following idea but I am not ...
1
vote
0answers
32 views

Finding the $\text {Im } (f^2)$

Let $f: \mathbb{R}^3 \to \mathbb{R}^3$ defined by $f(a,b,c)=(c-b,a-c,b-a)$ be a linear application. The matrix of $f$ is $A=\begin{bmatrix} 0 & -1 &1 \\ 1&0&-1\\-1&1&0 ...
0
votes
2answers
41 views

VERIFYING: Proving an $n \times n$ Matrix Vector Space

Let $V$ be the vector space made by the $n \times n$ square matrices. a) Prove that $S=\{A\in V|A^t=A\}$ is a subspace of $V$ b) Prove that $T=\{A\in V|A^t=-A\}$ is a subspace of $V$ c) Prove that ...
0
votes
1answer
24 views

Convergence of the maximum of a sequence of functions which converge uniformly on a closed interval

Can someone please verify this? Let $f_n$ be a sequence of continuous functions on a closed interval $I$ converging uniformly to $f$. Is it true that max $\{f_n(x):x\in I\}$ converges to max ...
4
votes
1answer
59 views

For what $\alpha$ does the series converge: $\sum^\infty_{n=2}\frac {1}{n^\alpha\log_2(n)}$

Let $\alpha\ge 0$ check for what $\alpha$ does the series converge: $$\displaystyle\sum^\infty_{n=2}\dfrac {1}{n^\alpha\log_2(n)}$$ I tried the condensation test and get: ...
2
votes
1answer
42 views

$a_{n+1}=a_n-a^2_n$ show the recursion sequence is convergent and find its limit

Let $a_1=\frac 2 3 , \ a_{n+1}=a_n-a^2_n$ for $n\ge 1$. Show the sequence is convergent and find its limit. In order to show convergence, I need to show that it's monotone and bounded. ...
1
vote
2answers
29 views

Proof Check: Every Cauchy Sequence is Bounded

Sorry if I keep asking for proof checks. I'll try to keep it to a minimum after this. I know this has a well-known proof. I understand that proof as well but I thought I'd do a proof that made sense ...
4
votes
1answer
417 views

Fake Proof of Prime Number Theorem

In David M. Burton's book on Elementary Number Theory I have found the following words, ... The first demonstrable progress toward comparing $\pi(x)$ with $\dfrac {x}{\ln x}$ was made by ... P. L. ...
1
vote
1answer
28 views

Algebra subgroup question

Let $G$ be a group, and let $H$ be a subgroup of $G$. Define $$C_G(H) := \lbrace g \in G \mid h \in H :gh=hg \rbrace.$$ (The set $C_G(G)$ is called the centralizer of $H$ in $G$.) Show that $C_G(H)$ ...
5
votes
5answers
258 views

Using dimensional analysis to evaluate $\frac{d}{dx}x^n$

Let $x$ have dimensions $[L]$ of length, so that $dx$ also has dimension $[L]$. Then $$\frac{d(x^n)}{dx}\;\text{has dimension}\;\frac{[L]^n}{[L]}=[L]^{n-1}.$$ Therefore $$\frac{d}{dx}x^n=cx^{n-1}$$ ...
1
vote
1answer
53 views

Proof Check: automorphism sends primitive root to primitive root

I was just wondering if this is a valid proof. I am assuming knowledge that if $\phi$ is an automorphism of a numeric field the $\phi$ fixes $\mathbb{Q}$. Also, if $\phi \in$ ...
1
vote
1answer
35 views

Let $G$ be a graph of girth $5$ for which all vertices have degree $\geq d$. Show that $G$ has at least $d^2+1$ vertices.

Could someone verify this? Pick a vertex $v$ of $G$. Pick distinct vertices $u_1, u_2, \ldots, u_d$ incident with $v$. Note that this can be done since $v$ has no loops and degree $\geq d$. For each ...
0
votes
2answers
23 views

Show that a connected graph on $n$ vertices is a tree if and only if it has $n-1$ edges.

Can someone please verify this? Show that a connected graph on $n$ vertices is a tree if and only if it has $n-1$ edges. $(\Rightarrow)$ If a tree $G$ has only $1$ vertex, it has $0$ edges. Now, ...
2
votes
1answer
51 views

If the integral of a non-negative function is $0$, then the function is $0$

Suppose that $f$ is a continuous function on $[a,b]$ and that $f(x)\geq0$ for all $x\in [a,b]$. Show that if $\int_a^bf(x)=0$, then $f(x)=0$ for all $x\in[a,b]$. Let $F(x)=\int_a^xf(x)$. Since ...
1
vote
0answers
19 views

Determining the number of solutions of the equation $y^2 = x^3 + x^2 + 5$ in $\mathbb{Z}/p^k \mathbb{Z}$ in function of $k$.

I found a question that asked me to discuss the number of solutions of the equation $y^2 = x^3 + x^2 + 5$ in $\mathbb{Z}/p^k \mathbb{Z}$ in function of $k$. I would like to use the multivariate ...
2
votes
1answer
40 views

Derivative of a definite integral (exercise)

Let, $$G(x)= \int_{x}^{\frac{1}{x}}\left (s+\frac{1}{s}\right)^9 ds$$ Find the derivative of $G(x)$. Here is my work: $$\frac{d}{dx}G(x)= \frac{d}{dx}\int_{x}^{\frac{1}{x}}\left ...
1
vote
1answer
43 views

If a group doesn't have subgroups of index 2 and 3, then any subgroup of index 4 is normal.

Let $G$ be this group and $H$ be any subgroup of index 4. $G$ acts on the set of left cosets of $H$ in $G$, which is a homomorphism $\varphi: G\to Aut(G/H) = S_4$. It is easy to see that $\ker ...
1
vote
1answer
50 views

How to prove a very basic algorithm by induction

I just studied proofs by induction on a math book here and everything is neat and funny: the general strategy is to assume the LHS to be true, and use it to prove the RHS (for the inductive step). Now ...
3
votes
2answers
117 views

Proving $\forall x\in\mathbb R : \dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$ with Cauchy's MVT

Prove for all $x\in\mathbb R$: $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$ Mclauren expansion: $$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+R_4(x)$$ ...
2
votes
3answers
96 views

Where does one use holomorphicity in the proof of Goursat's theorem?

Goursat's theorem: Let $f : U \to \mathbb{C}$ be a function that is holomorphic on the open set $U$. If $T$ is a triangle in $U$ and $\gamma$ is some smooth parametrization of that triangle, then ...
1
vote
1answer
31 views

How to show that $\sum_{n=1}^\infty (-1)^n\frac {x^2+n}{n^2}$ is uniformly convergent?

Show that $\sum_{n=1}^\infty (-1)^n\frac {x^2+n}{n^2}$ is uniformly convergent on arbitrary interval. I wanted to use M test for arbitrary [a,b] $|(-1)^n\frac ...
2
votes
1answer
33 views

Convergence of similar power series given a convergent series

Can someone verify this? Suppose that the series $$\sum\limits_{n=1}^\infty a_n x^n$$ has a radius of convergence $R$, where $0 < R < \infty$ (a) Find the radius of convergence of ...
5
votes
1answer
62 views

Uniform convergence of $\sum\limits_{n=1}^\infty \sin \left(\frac{x}{n^2}\right)$

Can someone please verify my answers? Consider the series $$\sum\limits_{n=1}^\infty \sin \left(\frac{x}{n^2}\right)$$ Prove that the series converges uniformly on the bounded interval $[-M, ...
0
votes
1answer
34 views

Problem regarding the roots of a quadratic equation

I came across this problem on the internet: If the roots of a quadratic equation $ax^2 + bx + b=0$ (where $a$ and $b$ are real numbers) are in the ratio $A:B$, then the value of ...
0
votes
1answer
42 views

Show that $(\log |x|)^2\notin \text{BMO}([-1,1])$.

I am trying to show that $u(x)\equiv (\log |x|)^2\notin \text{BMO}([-1, 1])$ by showing that it doesn't satisfy the John-Nirenberg inequality. If $u\in\text{BMO}[-1, 1])$ then this inequality says ...
3
votes
0answers
34 views

Divisibilty as relation set on $(\mathbb N \setminus \{0,1\})$

So i have to see if $\prec$ is order relation where two elements $(a,b)$ and $(c,d)$ are in relation $\prec$ if $a|c$ and $2b^{2}+6b\leq2d^{2} + 6 d$. This relation is defined on set $(\mathbb N ...
3
votes
1answer
64 views

About reparametrization of timelike curves in $\mathbb{L}^3$ (Lorentz-Minkowski space)

I think there is something wrong with the proof this text gives of Lemma $2.1.5$, in pages $19$ and $20$, for timelike curves. I used another function, and it seems to work. Either I'm wrong, or he ...
0
votes
1answer
47 views

The Weibull as the limiting distribution of the Burr distribution

I often deal with "payout patterns" which are vectors of the cumulative percentage of a loss that has been paid over time. For example, for $t \in [0, 1, 2, 3, 4, 5]$ I may have $p_t = (5\%, 15\%, ...
1
vote
2answers
52 views

Finding if a function with cases is differntiable on a point

Is $g$ differentiable on $x=0$ ? $$g(x)=\begin{cases}\dfrac{e^x-1}{x}&,x\neq0 \\ 1 &,x=0 \end{cases} $$ The derivative for $x\neq0$: $g'(x)=\dfrac{e^x(x-1)+1}{x^2}$, by taking the ...
1
vote
1answer
16 views

Prove $\{(x_1 ,x_2, 0) : x_1, x_2 ∈ F\}$ is a subspace of $F^3$.

$(x_1 ,x_2, 0) + (y_1, y_2, 0) =((x_1 + y_1), (x_2 + y_2), 0)).$ So, it's closed under +. $a(x_1 ,x_2, 0) = ax_1, ax_2, 0$. So, it's closed under *. Vector $(0, 0, 0) \in \mathbb F^3$ and its ...
1
vote
1answer
23 views

Prove that every subgraph of forest has at least one vertex of degree < 2

So I know that a forest is a graph that has no cycles. This is what I had in mind: Assume that we have the subgraph T, which has two options: to be connected or not. if it's connected it has to be a ...
0
votes
2answers
52 views

Existence of $\delta$

Exercise Assume that $K$ and $A$ are disjoint nonempty subsets of $ \Bbb{R}^n$ with $K$ compact and $A$ closed. Prove by using wat we know about $d(x,A)$ that there exists $\delta >0$ such that ...
0
votes
2answers
25 views

Proof of integral test

$f:[1,\infty)\rightarrow \mathbb{R}$ monotone decreasing and $f(x)\ge 0$ and $\int_1^{\infty} f(x) dx$ exists $\Rightarrow$ $\sum_{n=1}^{\infty} f(n)$ is convergent I need to prove this statement with ...
0
votes
0answers
25 views

Proving equivalence between basic feasible solution and vertex

I stumbled upon this proof of the Bertimas book on Linear optimization and I don't see what the "key ingredient" is that makes it work. Baxic feasible solution $\implies$Vertex Let $x^*$ be a ...
1
vote
4answers
42 views

Proof simplification

I am tasked with proving the following: $$\varnothing - A = \varnothing $$ My Attempt : Assume there exist $x \in $$\varnothing - A $ then $$ x \in \varnothing - A \Rightarrow x \in ...
0
votes
1answer
31 views

Can someone criticise my incorrect proof about a set being open?

In the question I have to decide whether the set $S=\{(x,y)\in\mathbb{R}^2\;|\;x/y\leq 7\}$ is open, closed or neither. I attempted to prove it was closed but it turns out it is neither can someone ...
1
vote
0answers
21 views

Subgroups of SO$(2)$

I'm doing an independent study with John Stillwell's Naive Lie Theory and I wanted to know if I'm on the right track. I'm just looking for some confirmation that these are acceptable answers. Find ...
1
vote
0answers
32 views

If $f(x)$ is integrable on $[a,b]$ then $c\cdot f(x)$ is also integrable and $\int_a^b c\cdot f(x) dx=c\cdot \int_a^b f(x) dx$

I proved the first part of this theorem which says that $c\cdot f(x)$ is integrable,but how to prove that $\int_a^b c\cdot f(x) dx=c\cdot \int_a^b f(x) dx$? Maybe it provides a bit help if i tell how ...
1
vote
3answers
30 views

Verification of Proof strategy

I am tasked with proving the following : $$A \cap B^c \subseteq (A \cap B)^c$$ I came up with the idea of using a combination of De Morgan's laws, rule simplification and rule of addition to prove ...
0
votes
3answers
34 views

If $Y$ is compact and $f : X \rightarrow Y$ is a map whose graph $G = \{ (x,f(x) : x \in X\}$ is closed in $X \times Y$ , then $f$ is continuous.

If $Y$ is compact and $f : X \rightarrow Y$ is a map whose graph $G = \{ (x,f(x) : x \in X\}$ is closed in $X \times Y$ , then $f$ is continuous. Let $C \subseteq Y$ be a closed. Let $x \in X - ...
2
votes
1answer
34 views

$\varepsilon$-$\delta$ proof of continuity of floor function $\lfloor x\rfloor$

I would just like to ask someone to confirm or correct the following 'proof' of continuity of the floor function. Let $\varepsilon>0$ be given. Set $\delta:=\min\lbrace x-\lfloor x\rfloor,\lceil ...
2
votes
2answers
55 views

Proving $a_n=\frac 1 2 \max\{a_{n-k},a_{n-k+1},…,a_{n-1}\}+1$ is monotone and finding its limit

Let $(a_n)^{\infty}_{n=1}$ defined like so: let $2\le k\in \mathbb N$ and $a_1,a_2...a_k\in \mathbb R$, $a_j\le 2, \ j=1,2,...,k$. Let $\forall n\ge k+1 : a_n=\frac 1 2 ...
0
votes
3answers
28 views

Differentiability of $f(x) = x \sin \frac{1}{x}$ for $x \neq 0$ and $0$ for $x = 0$.

Can someone please verify this (I admit the proof is very terse, but is the reasoning correct)? Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be defined by $f(x) = x \sin \frac{1}{x}$ for $x \neq ...
0
votes
2answers
27 views

Is this proof rigorous enough? Subspace of discrete space

Problem: Every subspace of a discrete space is discrete. Proof1 :Let $X$ be a discrete space with the discrete topology $\tau = 2^X$ and $Y$ be subspace with its topology $$\tau_Y = \{ Y \cap U : ...
0
votes
2answers
76 views

Proof that $\int_0^b \int_0^x f(t) dt dx = \int_0^b x f(x) dx$

I have the following statement to prove/disprove: If $f$ is a continuous function in the interval $[0,b]$ using the fundamental theorem of calculus and integration by parts, we have ...
4
votes
0answers
60 views

Showing that $|x-y| \leq |x| +|y|$ for $x.y \in \mathbb{R}$.

I know from intuition that $|x-y| \leq |x| +|y|$ for $x.y \in \mathbb{R}$. The way I would prove it is to use the triangle inequality: $|x-y| = |x+(-y)| \leq |x| +|-y| = |x|+|y|$ for $x.y \in ...
1
vote
3answers
68 views

Is this proof legal?

Let $\left(P,\le\right)$ denote a poset. Statement: if every sequence $p_{1}\leq p_{2}\leq\cdots$ in $P$ stabilizes (in the sense that for some $n$ we have $k>n\Rightarrow p_k=p_n$) then every ...
3
votes
0answers
83 views

Countable transitive $T \vDash ZFC-P$ with $\approx$ not absolute: why do we need $H(\aleph_3)$?

I am trying to solve Exercise IV.3.31 from Kunen's Foundations of Mathematics. I think I have a solution but I am confused by one of the hints. By request, here is the text of the exercise. ...
0
votes
1answer
41 views

Rudin: A compact metric space $K$ has a countable base, therefore $K$ is separable.

Hi this is a problem from Rudin's Princ. of Mathematics. I was hoping someone could check this part of my proof for the following question, comments would be very appreciated!: $25.$ Prove that ...