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

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3
votes
1answer
47 views

Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$ [duplicate]

Can someone please verify this? Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$ let $x \in \mathcal{P}(B)$. Then, $x \subseteq B$ This implies that $$\forall a \in ...
1
vote
1answer
24 views

Proof of isometries and inverses on the plane

I am taking a course on Intuitive Geometry. I am quite new to intuitive proofs however feel I've done pretty well thus far. Here is my theorem: Prove: That every isometry has an inverse. $Proof.$ ...
1
vote
2answers
63 views

Proof Verification for Homework

If $n$ is odd, then $n^2$ is odd. $1$) $n = 2k + 1$ (Definition of an odd number) $2$) $n^2 = (2k+1)^2 = (2k+1)(2k+1) = 4k^2 + 4k + 1$ (Distributive Property) $3$) $4k^2 + 4k + 1 = 2(2k^2 + 2k) + ...
2
votes
1answer
102 views

Proving $\displaystyle \int_0^{1} \sin\left(x + \frac{1}{x}\right)\, dx$ Exists

As the title says, I need to show that $$\int_0^{1} \sin\left(x + \frac{1}{x}\right)\, dx$$ exists. After performing the substitution $x = 1/u, dx = -1/u^2 du$, the integral becomes ...
0
votes
2answers
80 views

Is this NOT considered a proof?

Define the boundary of $A$ as $$Bd(A) = cl(A) \cap cl(X - A).$$ Show that $cl(A) = int(A) \cup Bd(A)$. The solution tries to show that $cl(A) \subset cl(A).$I thought I would do it directly, $int ...
2
votes
1answer
42 views

Answer check on two series

I want to determine if these two are absolutely convergent, conditionally convergent or simply divergent. 1) $$\sum_{n=2}^\infty \left(\frac xn - \frac x{n-1}\right)$$ $$= \frac x2 - \frac x1 + ...
6
votes
1answer
124 views

Limit related to the recursion $a_{n+1}=(1-n^{-1/4}a_n)a_n$

Let $0<a_{1}<1$, with $$a_{n+1}=a_{n}(1-n^{-\frac{1}{4}}a_{n})$$ Does there a real numbers $A$ which make the limit $$ ...
4
votes
2answers
80 views

Error in Hungerford's algebra proof? Left id & inv = group

Prop 1.3 in Hungerford's Algebra said that if $G$ is a semigroup and there exist a left identity and each element have a left inverse, then $G$ is a group. The proof (and in fact, even the proposition ...
1
vote
1answer
57 views

Is it true , if $|A|=|B|$ and $|C|=|D|$, then $|A \times C| = |B \times D|$?

Check my proof, please. Divide into subsets $A \times C$ and $B \times D$ so that , all pairs with the same element belong to the same subset. Each such subset $|A \times C|$ bijective $C$, $|C|=|D|$ ...
0
votes
1answer
26 views

Direct Product of Torsion Subgroups

So I came up with this theorem while studying, and concocted a small proof, and I was wondering if someone could verify it, as I am very new to torsion groups/elements. I am open to all criticism. ...
2
votes
1answer
57 views

Proof that left adjoints preserve direct limits

I am reading Rotman's book on Homological algbra and have a slightly different proof of the statement in the title of this question. Am writing my attempt below. Could someone please advise me if I am ...
0
votes
2answers
60 views

A Proof of Legendre's Conjecture

http://vixra.org/pdf/1303.0048v1.pdf In the paper the author proposes an elementary proof of Legendre's Conjecture. I was wondering if the proof is correct, because till now, there is no accepted ...
0
votes
0answers
29 views

A planar graph has either 2 faces or 2 vertices of degree less than 3

Practicing for an upcoming test, I stumbled upon this question: A planar graph with at least three vertices has either 2 faces of length at most 3, or 2 vertices of degree at most 3. Which is a ...
1
vote
1answer
43 views

Check workings for Strong Induction (Proof by Contradiction)

I want to prove the following: Suppose that $P(n)$ is a statement involving a general positive integer $n$. Then $P(n)$ is true for all positive integers $n$ if: i) $P(1)$ is true, and ...
1
vote
1answer
16 views

Determining if any of these three are an ideal of $\mathbb{R}[x]$

$\mathbb{R}[x]$ denotes the ring of polynomials in $x$ with real coefficients. Let $I \subset \mathbb{R}[x]$ be the subset of those polynomials with constant coefficient $0$, and let $J \subset ...
1
vote
1answer
38 views

Proof of coset and normal subgroup

I have this question: Let $G$ be a group, $a,b\in G$ and let $H$ be a subgroup of $G$. i) Give the definition of the coset $aH$ ii) Prove that $aH = bH$ if and only if $a^{-1}b\in H$ ...
0
votes
2answers
47 views

If $g \circ f$ is injective, so is $g$

If $g \circ f$ is injective, so is $g$ I don't think this is true. I think that $f$ has to be surjective. So I am going to try to prove that: If $g \circ f$ is injective, and $f$ is ...
2
votes
3answers
72 views

$x^{1+\epsilon}$ is not uniformly continuous on $[0,\infty)$

There are two questions. First: is the proof underneath correct? Let $\epsilon>0$ and let $f(x)=x^{1+\epsilon}$. I aim to show that $f$ is not uniformly continuous on $[0,\infty)$. We will show ...
3
votes
2answers
37 views

Show $\cos\theta=\frac12(\text{tr}(g)-1)$ with $g\in\text{SO}(3)$

How can I show that for $g\in\text{SO}(3)$ given by $\begin{pmatrix}1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta\end{pmatrix}$ the equality ...
1
vote
1answer
23 views

Find all points where $f$ is differetiable

Let $$f(x) = \left\{ {\matrix{ {0,x \notin Q} \cr {{x^2}({x^2} - 1),x \in Q} \cr } } \right.$$ I already proofed that $f(x_0)$ is continuous iff $x_0\in \left\{0,1,-1\right\}$. Now, if $f$ ...
3
votes
2answers
58 views

Topology of test functions $\mathcal{D}(\mathbb R)$

(My motivation for the following question is to understand the distribution theory) The space of test functions: $\mathcal{D}(\mathbb R)= \{\phi:\mathbb R \to \mathbb R : \phi \in C^{\infty}(\mathbb ...
1
vote
1answer
34 views

Proving $M_p$ is maximal in $C[0,1]$

Let $M_p$ be the ideal of those continuous functions of $C[0,1]$ which have $p\in [0,1]$ as a zero. It is a commonly known fact that $M_p$ is a maximal ideal. However, the proof is generally ...
1
vote
2answers
43 views

Is there a direct proof of the following?

I have been warned by my Lecture as well as several other sources that while proof by contradiction is useful and is certainly needed in some cases, it is often overused. In a effort to learn, I ...
0
votes
0answers
25 views

How about integral version of Holder's inequality?

In light of the fact that Minkowski's inequality have integral version, I thought there might be one for Holder's as well. I cannot find any through searching (there is an infinite product version in ...
1
vote
1answer
64 views

How many connected components? (CSIR June'13)

Let $X= \{ (x,y)\in \mathbb{R}^2: x^2+y^2<5\}$ and K=$\{(x,y)\in \mathbb{R}^2: 1\le x^2+y^2\le2 \quad\text{or}\quad 3\le x^2+y^2\le 4\}$ Then, 1.$X\setminus K$ has three connected ...
0
votes
2answers
46 views

Is this a valid proof?

Q: Prove that $m^2 = n^2$ iff (if and only if) $m = n$ or $m = -n$ I began by assuming that the condition m = n or m = -n could be restated as |m| = |n|. Next, I rewrote that as $\sqrt{m^2} = ...
0
votes
0answers
33 views

Did I obtain this recursion in the Fourier domain correctly?

I would like to calculate the following recursion: $$f_n(x)=\int_{B}^{A}f_{n-1}(x-\omega)f(\omega)\mbox{d}\omega\quad\quad f_1(\omega):=f(\omega)$$ This is simply the convolution of $f$ with itself ...
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
37 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
74 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
86 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
69 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
44 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 ...