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

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4
votes
2answers
66 views

$f'$ is bounded and isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\lim_{x\to y}f'$ does not exist

Prove/disprove: $f$ has a bounded derivative and $f'$ isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\displaystyle\lim_{x\to y}f'$ does not exist. I think that if $f'$ ...
1
vote
0answers
29 views

Is this solution correct

Given $A+B=\frac{\pi}{4}$, find $(1+\tan A)(1+\tan B)$ My attempt: Since $\tan(A+B)=1=\frac{\tan(A)+\tan(B)}{1-\tan(A)\tan (B)}$, therefore $\tan(A)+\tan(B)+\tan(A)\tan(B)=1$, therefore, ...
4
votes
3answers
182 views

Integration by change the variable

Let, $\int_{-1}^1\sqrt{1+e^x}\operatorname{dx}$. Write as an integral of a rational function and compute it. Suggest: change the variable in order to eliminate the square root. My work was: ...
1
vote
3answers
148 views

Proposed proof of set theoretic result

I am tasked with proving the following: $$ (A - B)\cap (B-A) = \varnothing $$ My Attempt: Suppose there exist a $x \in (A - B)\cap (B-A) $ then: \begin{align*} x \in (A - B)\cap (B-A) &\iff ...
2
votes
0answers
38 views

Counter example of monotone union [duplicate]

I saw this exercise in "Elements of Abstract and Linear Algebra" by E. H. Connell: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, ...
4
votes
5answers
109 views

Showing a recursion sequence isn't bounded $a_{n+1}=a_n+\frac 1 {a_n}$

Show the sequence isn't bounded: $a_1=1$, $a_{n+1}=a_n+\frac 1 {a_n}$. Proof by contradiction: Let $M>0$ such that $\forall n: |a_n|< M$. Let $\epsilon >0 $ and for some $n=N, ...
2
votes
3answers
75 views

Proving $\frac2\pi x \le \sin x \le x$ for $x\in [0,\frac {\pi} 2]$

Prove $\frac2\pi x \le \sin x \le x$ for $x\in [0,\frac {\pi} 2]$. I tried to do this in two ways, I'm not sure about CMVT and I have a problem with the other way. Using Cauchy's MVT: RHS: ...
1
vote
2answers
49 views

Proving $ f(x)=(\frac {\sin x} {x})^{\frac {1} {x^2} }$ is uniformly continuous on $(0,1]$

Prove that $f(x)=\Large(\frac {\sin x} {x})^{\frac {1} {x^2} }$ is uniformly continuous on $(0,1]$. Basically what I need to show here is that there is a limit 'from the right' for $x=0$ so the ...
4
votes
3answers
87 views

Proving $\limsup\frac 1 {a_n}=\frac 1 {\liminf a_n}$ and $\limsup a_n\cdot \limsup \frac 1 {a_n} \ge 1$

Let $a_n$ be a sequence such that $\forall n\in \mathbb n: 0<a\le a_n\le b <\infty.$ Prove: $\displaystyle\limsup_{n\to\infty}\frac 1 {a_n}=\frac 1 ...
2
votes
3answers
70 views

Monotone Union of subgroups being subgroup

I saw this exercise in a book: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, if $\{H_t\}$ is a monotonic collection, show that ...
3
votes
1answer
35 views

Showing that for $s,t\in\mathbb{Q}$, we have $(s+t)^*= s^* + t^*$.

I'm working through the problems of Elementary Analysis Theory of Calculus, and for some reason, this question didn't make the solutions in the back of the book. I did a thorough search on Stack ...
0
votes
1answer
49 views

Is something wrong with a proof of $f(A) \cup f(B) \subseteq f(A \cup B)$?

Claim: $$f(A) \cup f(B) \subseteq f(A \cup B)$$ Suppose $$ y \in f(A) \cup f(B)$$ $$y \in f(A)$$ or $$y \in f(B)$$ $$\exists x_0 \in A (f(x_0) = y)$$ or $$\exists x_0 \in B (f(x_0) = y)$$ ...
2
votes
1answer
42 views

Generalized Induction Verification

Consider the following simple exercise. Prove or disprove: $\gcd(km, kn) = k \gcd(m, n)$, where $m, n, k$ are natural numbers. Now, this is easy to prove using prime factorization. Knowing that ...
3
votes
1answer
46 views

Can a proper Morse function $\mathbb{R}\to\mathbb{R}$ have infinitely many critical points?

Depending on interpretation, there may be an assumption missing from Exercise 6.1.4(a) in Liviu I. Nicolaescu's Invitation to Morse Theory: Suppose $f : \mathbb{R} → \mathbb{R}$ is a proper Morse ...
0
votes
3answers
47 views

Proving commutativity of addition for vector spaces

I'm trying to prove commutativity of addition for vector spaces, using the axioms for vector spaces. Apparently commutativity can be proven! Im having trouble getting a good feel for what is allowed ...
2
votes
1answer
39 views

$V$ is a linear space. Need to compute $T^n$

Given: $V$ is a linear space $(\dim V = n)$ and there is a linear transformation $T: V \rightarrow V$ that $T^n = 0$ and $T^{n-1} \ne 0$ , also there's $u \in V$ that $T^{n-1}(u) \ne 0$ Prove ...
1
vote
3answers
49 views

Proof for modulus via direct or contrapositive

I have to prove the following via direct proof or via contra positive. For $a,b\in \mathbb{Z} $ it follows that $ (a+b)^3 \equiv a^3 + b^3 \mod 3$ I'm unsure of the best way to approach this ...
0
votes
1answer
35 views

What is effective price of suger

The price of sugar was Rs$25.00$ on January. It got increased in February by 40%. In March the price was reduced by 40%. The new price will be: My solution: $25[1+0.4][1-0.4]=21$ Is it correct?
2
votes
2answers
69 views

$\exists x_0$ such that $f(f(x_0))=x_0$ prove that $f$ has a fixed point

Let $f:\mathbb R\to \mathbb R$ be coninuous. Suppose there exists $x_0$ such that $f(f(x_0))=x_0$. Prove that $f$ has a fixed point or in other words: $\exists c\in\mathbb R: f(c)=c$ . Suppose ...
2
votes
2answers
36 views

My version of order topology is Hausdorff

Can someone say something about my version of "order topology implies Hausdorff" (WLOG) Let $a <b$, and let $U_1,U_2$ be a neighborhood of $a,b$ respectively. Denote $U_1 = (a - \epsilon, a + ...
3
votes
1answer
53 views

Is this right? Topology with closures

I want to show that (possibly) $$cl(A-B) = cl(A) - cl(B).$$ I know that $$cl(A-B) \subset cl(A) - cl(B).$$ already, but for the other inclusion I tried this. Let $x \in cl(A) - cl(B)$, so that for ...
0
votes
1answer
46 views

$f$ is differentiable twice, bounded and has a minimum on $x_0$, prove that there's a point $y\in\mathbb R$ such that $f''(y)=0$

Suppose $f:\mathbb R\to \mathbb R$ is differentiable and there's a constant $c>0$ such that $f'(x)>c$ for all $x\in(a,\infty)$. Prove that $\displaystyle\lim_{x\to\infty}f(x)=+\infty$ ...
0
votes
3answers
59 views

Disproof of Gelfond-Schnieder Theorem [closed]

The Gelfond Schneider theorem somewhere says that "There exist 2 such irrational numbers a and b(where a doesn't equal to b), ab is rational. The solution is taken as (in many answers in stack ...
2
votes
4answers
67 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
32 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
418 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
259 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
119 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 ...