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

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1answer
20 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 ...
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0answers
14 views

On even almost perfect numbers other than the powers of two

(Note: There is a related question in MSE as well as in MO.) Let $\sigma(x)$ be the sum of the divisors of $x$. We say that $X$ is almost perfect if $\sigma(X) = 2X - 1$. Antalan and Tagle (in a ...
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0answers
22 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 ...
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0answers
10 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) ...
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0answers
31 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 ...
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1answer
30 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 ...
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1answer
46 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 ...
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1answer
30 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 ...
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1answer
23 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 ...
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1answer
32 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 ...
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4answers
138 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
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1answer
36 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
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1answer
46 views

Prove a complex function

Question: Show using the $\epsilon -\delta$ definition that
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3answers
41 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.
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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
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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 ...
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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 + ...
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1answer
51 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
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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 ...
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0answers
16 views

Lebesgue measure of region under curve

Let $(X,\Sigma,\mu)$ be a $\sigma$-finte measure space and $f \in L^+(X,\Sigma)$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$. Theorem: Define the area under the graph of $f$ to be ...
4
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1answer
35 views

if $r,s$ are rational numbers, then $r+s\sqrt2$ is irrational unless $s=0$?

if $r,s$ are rational numbers, Prove $r+s\sqrt2$ is irrational unless $s=0$? I need to prove this simple question, but not sure if my method is acceptable I'm trying to prove it by ...
0
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1answer
14 views

Prove that: $n^2+3n^3 + 6^{lgn} is $ $\theta(n^3)$

I'm asked to prove that: $n^2+3n^3 + 6^{lgn} is $ $\theta(n^3)$ I know that for Big O, I need to show: $f(n) <= c*g(n)$ But I'm not sure how to show this, since it involves theta. Any help would ...
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2answers
52 views

Prove for all integers n such that n ≥ 3, $ 4^3 + 4^4 + 4^5 … 4^n = \frac{4(4^n - 16)}{3}$

I am trying to prove this using mathematical induction, but I'm lost once I get to comparing the two sides of the equation. Proposition: For all integers n such that n ≥ 3, $ 4^3 + 4^4 + 4^5 … ...
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0answers
22 views

Riemann-Stieltjes Integral Substitution

I want to prove $\int^b_a\,f(g(x))\,dg(x) = \int^{g(b)}_{g(a)}\,f(x)\,dx$ for all f continuous. Firstly, $\int^b_a\,f(g(x))\,dg(x) = \int^b_a\,f(g(x))g'(x)\,dx$, since g is continuous and ...
1
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1answer
20 views

Do these non-homotopic maps induce the same map in reduced homology?

Consider two maps $f, g: X\to Y$, where $X=Y=\{ 0, 1 \}$ with discrete topology, $f$ is the identity and $g$ maps everything to 0. Then it's clear that $\widetilde{H}_0(X;\mathbb{Z})\cong \mathbb{Z}$ ...
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0answers
17 views

Verifying a startegy to prove convexity on partial domain

Assume you have the multivariate function $$f(x_1,x_2,..,x_n)$$ where: $x_i>0 \forall i$, and $\sum_i x_i = 1$. I need to show that $f$ is a convex function. My plan is to show that it is ...
0
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0answers
12 views

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

I want to show the following statement: If $u \in W^{1,p}(I) \bigcap C_c(I)$ 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 ...
1
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2answers
40 views

Find the polynomial $P$ of smallest degree with rational coefficients and leading coefficient $1$ such that $ P(49^{1/3}+7^{1/3})=4 $

Find the polynomial $P$ of smallest degree with rational coefficients and leading coefficient $1$ such that $$ P(49^{1/3}+7^{1/3})=4 $$ (Source:NYSML) My attempt Let $$ ...
2
votes
1answer
14 views

Loss of Dimension with Orthogonal Projection

I am working on trying to prove this claim: Consider a nonzero vector v in Rn. What is the dimension of the space of all vectors in Rn that are perpendicular to v? I think I intuitively understand ...
2
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2answers
29 views

Solvability of nilpotent groups

I'm uncertain about my proof about this exercise regarding nilpotent groups. If someone could me help me out, that would be appreciated. There's a post about this problem, but it uses another ...
0
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0answers
58 views

The condition that given polynomial is divisible by 3

In How can I prove that the following is divisible by 3?, I showed $k^3+3k^2+2k$ is divisible by $3$ using Euler's theorem, specifically, Fermat's little theorem. Then I thought that it is possible to ...
0
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1answer
24 views

For a sequence, why must $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {||x_n||} = 0$, or there exists a convergent subsequence with a nonzero limit?

Suppose I've got a sequence of vectors $\{x_n\}_{n∈N}$ in $\mathbb{R}^k$. Why is it that exactly one of the following three facts must hold: $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {x_n} = 0$, or ...
3
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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 ...
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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; $ $F_{(1)-1} = F^2_{(1)} + F^2_{(1)-1} ->$ $ ...
3
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4answers
101 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? $$\int_0^af(x)dx+\int_{f(0)}^{f(a)}f^{-1}(x)dx=af(a)$$
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2answers
129 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
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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
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1answer
29 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 ...
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2answers
37 views

How many equivalence relations over $\mathcal P(\mathbb N)$ satisfy: $[\{8\}]_S=\{A\in \mathcal P(\mathbb N)|A\neq \{1\}\wedge A\neq \{2\}\}$

How many equivalence relations $S$ over $\mathcal P(\mathbb N)$ satisfy: $$[\{8\}]_S=\{A\in \mathcal P(\mathbb N)\mid A\neq \{1\}\wedge A\neq \{2\}\}$$ Just to make sure I understand, the ...
0
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4answers
27 views

Proving $g$ is onto if $\{(f\circ g)(x)|x\in \mathbb R\}=\mathbb R$ and $f$ is injective

Let $f,g:\mathbb R \to \mathbb R$ and $\{(f\circ g)(x)|x\in \mathbb R\}=\mathbb R$. Prove that if $f$ is an injection then $g$ is a surjection. Since $f$ is an injection then $\exists ...
2
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1answer
38 views

Proving $\mathcal P(\mathbb N) \sim (\mathcal P(\mathbb N)\setminus \{\emptyset\})$

Prove: $\mathcal P(\mathbb N) \sim (\mathcal P(\mathbb N)\setminus \{\emptyset\})$ My attempt: One side is obvious since $\mathcal P(\mathbb N) \supset (\mathcal P(\mathbb N)\setminus ...
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2answers
267 views

Prove $|f|$ is constant implies $f$ is constant

Let $f$ be an entire function (differentiable everywhere over $\mathbb{C})$. Suppose that $|f|$ is constant. Prove that $f$ is constant. Hint: $|f|\equiv c$ implies that $u^2+v^2\equiv c^2$. Take ...
2
votes
6answers
238 views

How do we define arc length?

In trying to write a nice proof of the derivatives of $\sin(x)$ and $\cos(x)$, I encountered a serious problem, namely that I have never seen a proper definition of the notion of arc length. Based on ...
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0answers
23 views

Is it possible to construct the real plane from any line using only one vector on the line?

Suppose our space is the plane. Then any line on the plane is a subspace for that plane. Let's take $\vec v$ has a vector on the line, then the span of $\vec v$ is just the line itself. My prof ...
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1answer
38 views

Describing: $\{x\in \mathbb R\mid\forall y\left [(y\in \{t\in \mathbb N\mid t>3\})\to (y>x) \right ] \}$

Describe: $\{x\in \mathbb R\mid\forall y\left [(y\in \{t\in \mathbb N\mid t>3\})\to (y>x) \right ] \}$ with $x$ as the only parameter. The inner part can be rewritten: $(y\in \{t\in ...
0
votes
1answer
29 views

Proving $|S/R^2|=\aleph$ , $(x_1,x_2)S(y_1,y_2)\iff x_1^2-x_2=y^2_1-y_2$

Let $S$ be an equivalence relation over $\mathbb R^2$ such that: $(x_1,x_2)S(y_1,y_2)\iff x_1^2-x_2=y^2_1-y_2$ Prove that $|S/R^2|=\aleph$ One side is pretty simple: $|S/R^2|\le |\mathbb ...
2
votes
1answer
44 views

Prove Intersection of Large Enough Subsets is Non-empty

I am trying to prove the following claim. I did not find the claim in a book, but I believe it to be true. Claim: Consider a finite set $P\neq \emptyset$ and subsets $S_1, S_2, S_3 \subset P$, ...
1
vote
1answer
40 views

Proving $F(f)=\{x\in \mathbb R| f(x)=1\}$ isn't an injection

Let $$F: \mathbb N ^{\mathbb R}\to \mathcal P(\mathbb R)\\ F(f)=\{x\in \mathbb R| f(x)=1\}$$ Prove that $F$ isn't an injection. My attempt: Suppose $F(f_1)=F(f_2)$ and show that $f_1\neq ...
2
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0answers
61 views

Proving the Urysohn's metrization theorem by using the Nagata-Smirnov's metrization theorem

I need to prove the Urysohn's metrization theorem by using the Nagata-Smirnov's metrization theorem. Urysohn's metrization theorem: Every regular second-countable topological space is ...
2
votes
2answers
551 views

Least Upper Bound of 2 Sets

Let $A+B = \{a+b | a \in A, b \in B\}$ and denote the least upper bound of a set $X$ as $lub(X)$. Show that $lub(A+B) = lub(A) + lub(B)$. I'm finding this problem a little confusing. I've ...