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

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5
votes
1answer
19 views

Solution to $\frac{d}{d\frac{1}{x}} x$

If I want to solve $$\frac{d}{d\frac{1}{x}} x$$ is my approach correct? As $$\begin{align*} \frac{d}{d\frac{1}{x}}x&=\\ \text{with }\frac{1}{x}&=y\\ ...
2
votes
2answers
21 views

Show that if $\prod_\alpha X_\alpha$ is normal then so is $X_\alpha$.

Show that if $\prod_\alpha X_\alpha$ is normal then so is $X_\alpha$. This a question of proof-verification.So please suggest the required edits and fault in the logic but please don't give a ...
0
votes
2answers
22 views

$U(n)^2$ is a proper subgroup of $U(n)$

I'm trying to show that $U(n)^2$ is a proper subgroup of $U(n)$. Here $$ U(n)^2 = \{x^2 \mid x \in U(n)\}$$ where $U(n)$ is the group of units modulo $n$. My idea was to argue as follows: Consider ...
1
vote
0answers
10 views

Extension of co-coercivity in strongly convex functions

I am studying strongly convex functions and they mention if $f(x)$ is strongly convex with Lipschitz gradients $L$, which means $\parallel \nabla f(y) - \nabla f(x)\parallel \leq L\parallel x - y ...
1
vote
1answer
30 views

Intersection of two ideals

Let $A$ be a commutative ring and let $\mathfrak{a}$, $\mathfrak{b}$ be ideals in $A$. I am asked the following question: Show that $\mathfrak{a} \cap \mathfrak{b}$ is the largest ideal of $A$ ...
2
votes
4answers
54 views

Misconception of infinite prime numbers proof by contradiction?

I'm using the proof on this page, except with $q$ instead of $p$ on the left side. The misconception of the proof is that $q$ has to be a prime number. I found this using $n = 6$, which gets me $q = 1 ...
0
votes
0answers
11 views

$∀ Z^+$ can be written as $c_r * 3^r + c_{r-1} * 3^{r-1} + …. + c_2 * 3^2 + c_1 * 3 + c_0$

I am trying to prove the following statement: $∀ Z^+$ can be written as $c_r * 3^r + c_{r-1} * 3^{r-1} + …. + c_2 * 3^2 + c_1 * 3 + c_0$ where $c_r =$ 1 or 2, and $c_i$ = 0, 1, or 2 for all integers ...
0
votes
1answer
34 views

Any collection of n coins can be obtained using a combination of 3¢ and 5¢ coins where n ≥ 14

I am trying to prove this statement with strong induction, but I'm a little lost on the inductive step. Proposition: Let P(n) be the sentence ‘any collection of n coins can be obtained using a ...
0
votes
0answers
12 views

Degrees of vertices in a circuit must be even

Let $G$ be a graph with a circuit. Let $C$ denote the subgraph of $G$ consisting of vertices and edges of the circuit. Then for every vertex in $C$, $\deg (v)$ considered in $C$ is even. I would ...
0
votes
0answers
25 views

Does this method of finding the range of rational functions always work?

Consider the irreducible rational function in $\mathbb{R}^2$. $$y=\frac{A(x)}{B(x)}$$ where at least one term is quadratic and the other term has degree either 0, 1 or 2. The classic way of ...
0
votes
0answers
12 views

Can you analyze this identity involving the sum of divisors function and $rad(n)=\prod_{p\mid n}p$?

Let $\sigma(n)$ the sum of divisors function, then by Mobius inversion formula $$\sigma(n)=n-\sum_{\substack{d\mid n,d<n}}\sigma(d)\mu\left(\frac{n}{d}\right),$$ and since this function is ...
0
votes
1answer
27 views

Compute the shortest possible distance between a point on the graph of $f$ and a point on the graph of $f^{-1}$

Let $f(x)=(x+3)^2+\cfrac{9}{4}$ for $x\ge -3 $.Compute the shortest possible distance between a point on the graph of $f$ and a point on the graph of $f^{-1}$. My effort Let $P,Q$ be points on the ...
1
vote
1answer
35 views

The space of continuous functions on an interval has a countable dense subset and a countable basis

Give $\Bbb R^I$ the uniform metric, where $I = [0, 1]$. Let $C(I, \Bbb R)$ be the subspace consisting of continuous functions. Show that $C(I, \Bbb R)$ has a countable dense subset, and therefore a ...
2
votes
0answers
21 views

Number of elements of order $6$ in $\text{Aut}(\mathbb Z_{720})$

I tried to determine the number of elements of order $6$ in $\text{Aut}(\mathbb Z_{720})$. Please could someone tell me if this is correct? $$ \text{Aut}(\mathbb Z_{720}) \cong U(720) \cong U(9) ...
1
vote
0answers
37 views

Monoids and groups

everybody. I got this exercise from Jacobson. Let $M$ be a monoid generated by a set $S$ and suppose every element of $S$ is invertible. Show that $M$ is a group. Proof: every element of $M$ has ...
1
vote
1answer
35 views

Prove that the set $U = \{(123), (124), … , (12n)\}$ can be used to generate $A_n$.

A hint is provided with the proof prompt: $(abc) = (1ca)(1ab)$, $(1ab) = (1b2)(12a)(12b)$, and $(1b2) = (12b)^2$. My idea: $(1ab) = (12b)(12b)(12a)(12b)$. To solve for the other half of $(abc)$, I'm ...
0
votes
1answer
51 views

Proof that $\{\,\left]a,\infty\right [\mid a\in\mathbb{R}\,\}\cup\{\mathbb{R} \}\cup\{\emptyset \}$ is topology of $\mathbb{R}$

Proof that $\mathcal{T}:=\{\,\left]a,\infty\right [\mid a\in\mathbb{R}\,\}\cup\{\mathbb{R} \}\cup\{\emptyset \}$ is topology of $\mathbb{R}$. I have slight trouble on writing this down.. I'll first ...
0
votes
1answer
38 views

Two questions on Munkres -Topology

I have two questions: If $X$ is a countable product of spaces having countable dense subsets then does $X$ have a countable dense subset? Let $X$ $=\prod_{i=1}^\infty X_i$ .Let $D_i$ denote the ...
0
votes
1answer
29 views

Function is differentiable in all the points of its domain

I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function $\in C^k$ and a function is $C^k$ if is composition of ...
1
vote
1answer
35 views

Munkres Topology Article -30 Problem 5

Show that a metrizable space with a countable dense set has a countable basis. My try: Let $X$ be a metrizable space with a countable dense set $D$. Consider for each $n\in \Bbb ...
6
votes
4answers
140 views

Show that every compact metrizable space has a countable basis

Show that every compact metrizable space has a countable basis. My try: Let $X$ be a compact space and metrizable. Now for each $n\in \Bbb N$; I can consider the open cover $\{B(x,\frac{1}{n}):x\in ...
2
votes
1answer
38 views

Completeness of bounded linear maps

Let $X,Y$ be normed vector spaces over $\mathbb{C}$, and $L(X,Y)$ the space of all bounded linear maps from $X$ to $Y$. Its known that $L(X,Y)$ is a normed(operator norm) vector space. Theorem: ...
2
votes
1answer
52 views

Prove a complex function

Question: Show using the $\epsilon -\delta$ definition that
3
votes
3answers
42 views

Simplify $(k +1)! > (k + 1)^2$ to prove true for $k ≥ 4$

I am trying to prove this statement is true for $k ≥ 4$. I don't know how to work with $k + 1$ factorial, so I'm a little lost on proving this.
2
votes
3answers
53 views

Can I further simplify $5^k \cdot 5 + 9 < 6^k \cdot 6$ to prove this is true

I am trying to prove this statement, but I'm not sure where to go from here. Is don't think this is sufficiently reduced to conclude the statement is true, but I'm not positive. $k ≥ 2$ Can I ...
0
votes
3answers
47 views

Prove $n^3 + 7n + 3$ is divisible by 3 for all integers n ≥ 0

The statement I'm trying to prove is: $n^3 + 7n + 3$ is divisible by 3 for all integers n ≥ 0 I eventually need to prove $(k + 1)^3 + 7(k + 1) + 3$ is divisible by 3. I don't really understand ...
1
vote
2answers
26 views

Help Proving the Average is greater than B^(1/n)

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Define the following two numbers: $A = (a_1 + a_2 + \cdots + a_n) /n $ (The average of the numbers) $B = (a_1 + a_2 + \cdots + ...
1
vote
1answer
52 views

Show $ (\int_{-\infty}^\infty \sqrt{p}\sqrt{q}d\mu)^2\leq 2 \int_{-\infty}^\infty \min\{p,q\}d\mu $

Consider a random variable $X$ in $(\Omega, \mathcal{F}, \mathbb{P})$. Let $p,q$ be two densities with respect to a measure $\mu$ in $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where ...
0
votes
1answer
18 views

Continuity proving of function with delta-epsilon

Prove continuity of function with the delta-epsilon definition in point $x_o=0$ $$f:\mathbb{R}\rightarrow \mathbb{R}$$ $$f(x) = \begin{cases} x^2+1, & x \in \mathbb{Q} \\[2ex] 2^x, & x \in ...
0
votes
0answers
26 views

If $u \in W^{1,p}(I) \bigcap C_c(I)$ then $u \in W_{0}^{1,p}(I) $ where $ W^{1,p}(I)$ is the Sobolev Space

I want to show the following statement: If $u \in W^{1,p}(I) \cap C_c(I)$ then $u \in W_{0}^{1,p}(I) $. $W^{1,p}(I) $ is the Sobolev Space, i.e. the space consisting of the functions that are ...
0
votes
1answer
31 views

Prob. 9, Sec. 19 in Munkres' TOPOLOGY, 2nd edition: Equivalence of the choice axiom and non-emptyness of Cartesian product

The Axiom of Choice is as follows: Given a collection $\mathcal{A}$ of disjoint non-empty sets, there exists a set $C$ consisting of exactly one element from each element of $\mathcal{A}$; that ...
3
votes
0answers
52 views

Prove an annulus is homeomorphic to a cylinder

Let $A \subset \mathbb{R}^{2}$ be the annulus $A = \{(x,y) \in \mathbb{R}^{2} \colon 1 \leq x^{2} + y^{2} \leq 4 \}$. Prove that $A$ is homeomorphic to $S^{1} \times I$, where $I = [0,1]$ is the ...
1
vote
2answers
36 views

If $A$ and $B$ are conneted and $A\cap B\neq \emptyset$, then $A\cup B$ is connected

Can you please let me know if my proof is reasonable? Prove: If $A$ and $B$ are conneted in $\mathbb{R}^n$ and $A\cap B\neq \emptyset$, then $A\cup B$ is connected Proof: Suppose that $A\cap B$ is ...
0
votes
0answers
17 views

Is my proof correct? Convex optimization

There's a theorem that says that if $C \subset \mathbb R^n$ is a convex set, then $x^* \in C$ is the closest point in $C$ to $y \notin C$ if and only if $(y-x^*)\cdot(x-x^*)\leq 0$ for all $x \in C$. ...
1
vote
0answers
37 views

Wrong result: a continuous function has zero $p$-variation, for every $p$. Where's the error?

Let $\Pi_n$ be a sequence of partitions with $|\Pi_n| \to 0$. Then the $p$-variation of a continuous function $g$ along the partitions $\Pi_n$ is defined as $$V_T^p(g) = \lim_{n \to \infty} V_T^p(g, ...
1
vote
0answers
24 views

How to make inductive step for a Fibonacci proof [duplicate]

I have to prove $F^2_{n−1} = F^2_n + F^2_{n−1}$ for any $n >=1$ by induction (for the Fibonacci sequence). For the basis step, I have: $n = 1; $ $F_{(1)-1} = F^2_{(1)} + F^2_{(1)-1} ->$ $ ...
0
votes
1answer
70 views

Prove 1 is not the largest integer?

This proof looks extremely flawed, but I'm new to proofs so I'm not completely sure what is allowed and what isn't. Here it is: Let $n$ be the largest positive integer. Then $n$ must be $\geq 1$. ...
1
vote
2answers
56 views

Prove $\frac{2ab}{a+b}\leq\sqrt {ab}$

$a$ and $b$ are both positive real numbers. I'm supposed to work backwards (i.e. start with what I'm trying to prove and change it until something is absolutely true, then start from what is ...
1
vote
0answers
40 views

Limits of two fixed points of $E_\mu(x) = \mu e^x$

Please let me know if this proof is OK. Problem statement: Given that $E_\mu(x) = \mu e^x$, where $0 < \mu < 1/e$, show that if $q_\mu < p_\mu$ are fixed points, where $q_\mu$ is attractive ...
1
vote
0answers
17 views

Verification on classification of singularities

In an exercise, I'm asked to classify the singularities of these functions: $\qquad i) f(z)=\frac{1}{(z-1)^2} \qquad ii)f(z)=\frac{1-\cos z}{z^2} \qquad iii) f(z)=\frac{z^2-1}{z-1}$ I don't know why ...
3
votes
4answers
103 views

The sum of integrals of a function and its inverse: $\int_{0}^{a}f+\int_{0}^{f(a)}f^{-1}=af(a)$

Regarding real numbers, the following appears to be true, or at least true with some modifications. Could you help me for the proof? $$\int_0^af(x)dx+\int_{f(0)}^{f(a)}f^{-1}(x)dx=af(a)$$
12
votes
2answers
130 views

Prove $\int_{0}^{x}f+\int_{0}^{f(x)}f^{-1}=xf(x)\qquad\text{for all $x\geq0$}$ [duplicate]

Suppose that the function $f:[0,\infty)\rightarrow\mathbb{R}$ is continuous and strictly increasing and that $f:(0,\infty)\rightarrow\mathbb{R}$ is differentiable. Moreover, assume $f(0)=0$. ...
1
vote
1answer
31 views

$E\backslash C$ is relatively open in $E$

If $E \subset \mathbb R^n$ and $C \subset E$ is relatively closed in $E$, then $E\backslash C$ is relatively open in $E$.
2
votes
1answer
32 views

Verification of proof of propositional logic

I made a proof for the following theorem. But I'm not completely certain that it's fully correct. Suppose $\phi$ is a propositional formula and that the two evaluations $v$ and $w$ are equal for ...
2
votes
2answers
35 views

uniqueness of solution of diff equation second order

In the case of a diff equation where $r_1=r_2$ the solution to $y''+py'+qy=0$ is $y=Ce^{r_1 x}+Dxe^{r_1 x}$ (I) In a proof I have for uniqueness of solution they start by defining $y=u e^{r_1 x}$ ...
2
votes
2answers
55 views

Let R be a commutative ring, and let P be a prime ideal of R. Suppose that P has no nontrivial zero divisors in it. Show that R is an integral domain.

Let R be a commutative ring, and let P be a prime ideal of R. Suppose that P has no nontrivial zero divisors in it. Show that R is an integral domain. My proof: Take $r,s,a \in R$ with $ar = ...
-2
votes
1answer
51 views

The Zeros Localization Theorem and the Extreme value Theorem

My intro to analysis book calls the following theorem The Zeros localization theorem: Theorem: Let $p(x)=x^n + a_{n-1}x^{n-1} +\dots+a_1 +a_0,\ x \in \mathbb R$ , be a polynomial. Then all the zeros ...
2
votes
1answer
64 views

Is my proof valid? Integration of logarithmic function.

After this question: ...
0
votes
0answers
23 views

Proving $\forall f\in \mathbb R ^{\mathbb R}(f\neq i_{\mathbb R})\to (\exists g\in \mathbb R ^{\mathbb R} (f\circ g \neq g\circ f))$

Prove $\forall f\in \mathbb R ^{\mathbb R}(f\neq i_{\mathbb R})\to (\exists g\in \mathbb R ^{\mathbb R} (f\circ g \neq g\circ f))$ Proof by contradiction: For all $f(x)\neq x$ and $(\forall g\in ...
1
vote
1answer
33 views

Is this a correct to prove that there are no solutions for these trigonometric functions?

This is part of a problem that I've been doing; it turns out the way I was doing it was wrong but I've still got a question about one method I used in trying to prove the problem. I ended up trying to ...