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

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0
votes
1answer
42 views

Proof of sum-free set in $\mathbb{Z}_p$

Consider $a \in \mathbb{Z}_p \backslash\{0\}$ and define $aS=\{as | s \in S \}$. I want to show that $S$ sum-free over $\mathbb{Z}_p \iff aS$ sum-free over $\mathbb{Z}_p$, and then I want to show that ...
2
votes
1answer
33 views

Is it possible to have $|f(x) - f(y)| \leq M\| x - y \|$ under such conditions?

Let $f: A \to \mathbb{R}$ be differentiable on an open convex $A \subset \mathbb{R}^{n}.$ If $\| \nabla f \| \leq M$ on $A$ for some $M > 0,$ is it possible to have $$|f(x) - f(y)| \leq M \| x - y ...
2
votes
2answers
252 views

how to prove that the following is not a regular language?

the language we want to disprove is : $$ L = \{ 0^i1^j| gcd(i,j)=1 \} $$ my attempt : i used the pumping lemma this way: consider the set of strings of the form $0^p1^q$ such that $n <=p$ and ...
4
votes
2answers
97 views

How much does Proof writing improve over the years?

This is a very soft question. Just a bit of background: I'm a junior in high school taking Analysis I and II out of Baby Rudin at a very well-recognized university. I find quite a few of his ...
3
votes
2answers
111 views

Prove $f_n\to f$ on $[a,b]\implies \int_a^b|f_n-f|\to 0$

Suppose $f,f_n$ are measurable and uniformly bounded on $[a,b]$. Prove $f_n\to f$ on $[a,b]\implies \int_a^b|f_n-f|\to 0$ Attempt: We note that since $f$ and $f_n$ are bounded and are ...
3
votes
0answers
152 views

An argument for “Brocard's problem has finite solution”

Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard%27s_problem. According to Brocard's problem ...
0
votes
2answers
35 views

Proof infinity=1 IDK is there an error?[Solved]

Can anyone find the error in this? Or is this just another divergent series?
1
vote
1answer
15 views

Prove that $(A,B)\sim(P,Q)$ and $(C,D)\sim (P,Q)\implies (A,B)\sim (C,D)$?

I have the following laws: And I did the following: $(A,B)\sim(P,Q)\wedge (C,D)\sim (P,Q) \stackrel{?}{\implies} (A,B)\sim (C,D)$ $(A,B)\sim(P,Q)\wedge \stackrel{symmetry}{(P,Q)\sim ...
2
votes
2answers
40 views

Proving that for Fibonacci numbers $a_n \lt (\frac {1 + \sqrt 5} 2)^n$ for $n \ge 1$

I'd like to prove that for Fibonacci numbers $a_n \lt \left(\frac {1 + \sqrt 5} 2\right)^n$ for $n \ge 1$. I suppose it needs induction so, after verifying the trivial case $n=1$, the inductive step ...
2
votes
0answers
28 views

Generalization of Dirichlet convolution

The Wikipedia page on the Mobius inversion formula gives the following formula in passing: if $$G(x)=\sum_{k=1}^x \alpha(x)F(x/k)$$ for some arithmetic function $\alpha(n)$ possessing a Dirichlet ...
-1
votes
1answer
20 views

finite dimensional vector spaces with a symmetric bilinear form have an orthogonal basis

I am confused about case 2, where $\exists v \in V$ such that $f(v,v) \not= 0$. I have some questions: 1) where in the proof have we used the fact that $f(v,v) \not= 0$ - I don't see how this was ...
1
vote
2answers
45 views

Prime Ideals and multiplicative sets

I am currently studying a course on commutative algebra and came across this statement: An Ideal $I$ in a ring $R$ is prime if and only if $R\setminus I$ is a multiplicative set. I have proved ...
3
votes
2answers
68 views

Every local property for $\mathbb{R}$ (any Connected Separable Space) holds globally?

I'm given this problem : Prove that "being polynomial" is a local property, meaning if $f: ℝ → ℝ$ is a polynomial in a neighborhood of each real point, then $f$ is a polynomial. I think I ...
0
votes
1answer
75 views

$3 \times 3$ matrices are similar if and only if they have the same characteristic and minimal polynomial

I want to prove: $B$ is similar to $A \Leftrightarrow m_A(x) = m_B(x)$ and $P_A(x) = P_B(x)$, where $m,P$ are the minimal and characteristic polynomial, respectively. "$\Rightarrow$" Let $A$ to ...
2
votes
4answers
49 views

Finding what $p$ does the series: $\sum_{n=1}^{\infty}\frac{\ln n}{n^p}$ converges

For what $p$ does the series: $\displaystyle\sum_{n=1}^{\infty}\frac{\ln n}{n^p}$ converges? My attempt: I wanted to use the limit comparison test and compare it with $\frac 1 {n^p}$ but it ...
1
vote
1answer
45 views

In probability, what does P(A-B) stand for?

I have the next problem: Prove that $\forall A, B \subseteq \Omega, P(A-B) \ge P(A)-P(B)$ But, what does $P(A-B)$ mean? I thought it could mean the probability of all the events in $A$ that are not ...
1
vote
0answers
33 views

Proof verification: any n-th order complex polynomial has at most n distinct roots

Here is a proof I came up with in the exam I just took. But I suspect there may be some issues since I think it seems too simple. Proof Let $p_n(x)$ denote a complex polynomial of order $n$ ...
5
votes
0answers
230 views

For what values of t, the solution for this equation exist

I need help in finding maximal solution for the problem: $$ \cases {{\dot{x} = x^2+t}\\{x(0)=0}}$$ I know that because $x(1) \geq \frac{1}{2}$ and that every solution $x(t)$ of the problem is greater ...
1
vote
7answers
122 views

Eight points are in/on the circle of radius 1cm. Show that distance between some two points is less than 1cm.

Original problem If 8 points in a plane are chosen to lie on or inside a circle of diameter 2cm then show that the distance between some two points will be less than 1cm. My proof Let the points be ...
2
votes
1answer
40 views

A basic question regarding Lebesgue's density theorem

Here is the question from Pugh's Real Mathematical Analysis: My answer to $b)$ is that for a closed square, points on corner has density $1/4$, while on the sides the density is $1/2$. But how to ...
8
votes
1answer
46 views

Let $H$ have order $m$ and $K$ have order $n$, where $m$ and $n$ are relatively prime. Then $H \cap K=\{e\}$

Let $H$ and $K$ be subgroups of $G$. Let $H$ have order $m$ and $K$ have order $n$, where $m$ and $n$ are relatively prime. Then $H \cap K=\{e\}$ My proof: Let $H$ and $K$ be subgroups where ...
3
votes
1answer
47 views

Prove that $R$ is an equivalence on $\mathscr P(A)$. Is this correct?

Suppose $B\subseteq A$, and define a relation R on $\mathscr{P}(A)$ as follows: $$R=\{(X,Y)\in\mathscr{P}(A) \times \mathscr{P}(A)\mid(X\mathrel{\triangle} Y)\subseteq B\}$$ Prove that $R$ is an ...
4
votes
3answers
293 views

What is the mistake in this proof?

During a long night without sleep I managed to come up with a proof for a statement I know is false, and for the life of me I cannot figure out what I did wrong. Where is my mistake? Theorem: Let ...
0
votes
0answers
17 views

Deeply confused while trying to understand the derivation of infinite product representation of gamma function

I am trying to understand how in the world Euler figured out the infinite product representation of Gamma function. $$\Gamma (z)=\lim_{n\rightarrow \infty }\frac{n!n^{z}}{z(z+1)\cdots(z+n)}$$ Of ...
2
votes
0answers
19 views

Easy property of projection operators

Quick sanity check: If $V$ is a finite-dimensional $K$-vector space and $\pi:V \to V$ is a projection (i.e. an idempotent $K$-linear map), then $\text{tr}(\pi)=\dim_K (\text{im}(\pi)).$ Proof: As ...
5
votes
1answer
39 views

Function on a Power Set

Let $f\colon \mathcal{P}(A)\mapsto \mathcal{P}(A)$ be a function such that $U \subseteq V$ implies $f(U) \subseteq f(V)$ for every $U, V \in \mathcal{P}(A)$. Show there exists a $W \in \mathcal{P}(A)$ ...
0
votes
2answers
50 views

$\|(g\widehat{(f|f|^{2})})^{\vee}\|_{L^{2}} \leq C \|f\|_{L^{2}}^{r} \|(g\hat{f})^{\vee}\|_{L^{2}}$ for some $r\geq 1$?

Let $g\in C_{c}^{\infty} (\mathbb R)$, and $f, |f|^{2}f\in L^{2}(\mathbb R)\cap C_{0}(\mathbb R)$ (where $C_{c}(\mathbb R)$ is the class of smooth functions with compact support and $C_{0}(\mathbb R)$ ...
1
vote
0answers
64 views

Sum of geometric series

Let's say I have the series: $1+(x+1)+(x+1)^2….$ if $|x+1|<1$, what is the sum of infinite geometric series? This is my thinking: I have the formula $S= a \dfrac{1-r^n}{1-r}$ Now we know that ...
2
votes
1answer
54 views

Can you verify this proof of the Schroeder-Bernstein theorem?

I'm a freshman in college and my professor challenged us to find a proof of this theorem. Please don't give me the answer but please verify if this proof works or, if not, if it is the start of a ...
1
vote
3answers
38 views

Proving gcd($a,b$)lcm($a,b$) = $|ab|$

I was trying to prove that $$ dm = |ab|$$ where $d$ = gcd(a,b) and m = lcm(a,b). So I went about by saying that $a = p_1p_2...p_n$ where each $p_n$ is a prime. Same applies to $b = q_1q_2 ... q_c$. ...
2
votes
4answers
72 views

Prove if $7\mid a^2+b^2 \longrightarrow 7\mid a$ and $ 7\mid b$

Prove if $7\mid a^2+b^2 \longrightarrow 7\mid a$ and $7\mid b$ What I did: I found the possible remainders for $a^2$ are $0, 1, 2$ and $4$. I think I should say $r_7(a^2)+r_7(b^2)$ can't equal any ...
7
votes
0answers
70 views

Is a compact, simply-connected 3-manifold necessarily $S^3$ with $B^3$'s removed?

Let $M$ be a compact, simply-connected 3-manifold (which is also smooth and connected). Is $M$ diffeomorphic to $S^3$ with a finite number of $B^3$'s removed? This seems like a handy fact, but I ...
0
votes
2answers
60 views

Show that $\mathbb{Z}$ and $2\mathbb{Z}$ are not isomorphic as rings.

Show that $\mathbb{Z}$ and $2\mathbb{Z}$ are not isomorphic as rings. My attempt: Suppose $\mathbb{Z}$ and $2\mathbb{Z}$ are isomorphic as rings, Let $\phi: \mathbb{Z} \rightarrow 2\mathbb{Z}$ be the ...
1
vote
2answers
48 views

If both $f$ and $g$ are not integrable, then $f+g$ and $fg$ are not integrable

If both $f$ and $g$ are not integrable, then $f+g$ is not integrable I think this is false. Take $f(x) = \begin{cases} 1 & \text{ if } x \in \mathbb{Q} \\ 0 & \text{ if } x \in \mathbb{Q}^c ...
6
votes
2answers
254 views

RH would follow from $\displaystyle \frac{p_{n+1}}{p_{n+1}-1}<\frac{\log\log N_{n+1}}{\log\log N_n} $ for all $n>1$; what is my mistake?

Let $N_n=\prod_{k=1}^np_k$ be the primorial of order $n$,$\gamma$ be the Euler-Mascheroni constant and $\varphi$ denote the Euler phi function. Nicolas showed that if the Riemann Hypothesis is true, ...
3
votes
5answers
146 views

Prove that $2^n\le n!$ for all $n \in \mathbb{N},n\ge4$

The problem i have is: Prove that $2^n\le n!$ for all $n \in \mathbb{N},n\ge4$ Ive been trying to use different examples of similar problems like at: ...
0
votes
2answers
22 views

Short proofs about integrability

If true, the prove it; if false, the provide a counterexample. a) If $f$ is integrable, but $g$ isn't, then $f + g$ is not integrable. True: Assume that $f + g$ is integrable, then $f$ and $g$ must ...
1
vote
2answers
43 views

If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$

If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) dx = 0$ iff $f(x) = 0$ "$\Rightarrow$" Assume by contradiction that $f(x) \neq 0$ for some $x_0 \in [a, b]$. Without loss ...
0
votes
1answer
35 views

How many strictly increasing functions can be formed?

Let $A=\{x \in\mathbb{N}~|~x\leq10\}$, $B=\{x \in\mathbb{N}~|~x\leq100\}$ $f: A \longrightarrow B$ How many strictly increasing functions can be made? I thought: I have $91$ options for $f(1)$ ...
0
votes
1answer
35 views

Finite Group and Conjugacy Class Question

Let $G$ be a finite group. Show that $[G:Z(G)]$ cannot be prime. Assume $G$ is abelian. The $Z(G) = G$ and $[G:Z(G)] = 1$, not a prime. Now consider the case where $G$ is not abelian. If ...
0
votes
1answer
17 views

Some issue understanding the proof that a limit is unique

We want to show that, if $lim(s_n) = s$ and $lim(s_n) = t$, then $s = t$. The definition of limit $a$ for a sequence $s_n$ is: For each $\epsilon > 0$, there exists an $N$, such that ...
2
votes
2answers
37 views

Show that $\widehat{f}(n)$ is zero for odd $n$

The following problem is from Stein´s Introduction to Fourier analysis: Suppose that $f(\theta + \pi)=f(\theta)$ for all $\theta \in \mathbb{R}$ Show that $\widehat{f}(n)$ is zero for odd $n$. My ...
4
votes
1answer
98 views

Existence of bijection that reorders elements?

Suppose I have some function $f:\mathbb{R}\to[0,1]$. Does there necessarily exist a bijective mapping $g:\mathbb{R}\to\mathbb{R}$ such that $g(x)\leq g(y)$ implies $f(x)≤f(y)$? If not, does it help if ...
4
votes
3answers
90 views

proving that $\mathbb{Q}(\sqrt{5}, \sqrt{6}) = \mathbb{Q}(\sqrt{5}+ \sqrt{6}) $

Here is an extract from my Galois Theory notes proving that $\mathbb{Q}(\sqrt{5}, \sqrt{6}) = \mathbb{Q}(\sqrt{5}+ \sqrt{6}) $ My question is after rearranging equation (1) has my lecturer omitted an ...
0
votes
0answers
25 views

Let $R$ be a UFD, and let $f(X) \in R[X]$ be a primitive polynomial. Prove that if $g(X) | f(X)$, then $g(X)$ is primitive.

Let $R$ be a UFD, and let $f(X) \in R[X]$ be a primitive polynomial. Prove that if $g(X) | f(X) \in R[X]$, then $g(X)$ is primitive. Attempted Proof: Since $f(x)$ is primitive, its coefficients do ...
4
votes
1answer
65 views

Fake proof: Equivalence of norms

Good morning. I'm having a hard time finding what's wrong with the following argument. Let $f$ be any function in $C^{1}([0;1])$ and let $||f||$ and $N(f)$ be two norms defined as follows: $$||f|| = ...
1
vote
3answers
16 views

Prove $\sup S \leq \inf T$, if $s \leq t$, $\forall s \in S$ and $\forall t \in T$

I have the following exercise: Prove $\sup S \leq \inf T$, if $s \leq t$, forall $s \in S$ and $t \in T$. Note that $S$ is bounded above and $T$ is bounded below. This might seem too obvious, ...
-1
votes
0answers
604 views

$ (x+y) \geq (p_n +2) $?

I recently worked on a previous idea of mine (a prime number inequality, which I had posted in this community but didn't know Latex then and couldn't discuss it's proof). I was wondering how powerful ...
1
vote
0answers
25 views

Verifying proof that $f$ has an inflection point at zero if $f$ is a function that satisfies a given set of hypotheses.

Prove that if $f$ is a function $f(x): f'(x) > 0$ $\forall x: x \neq 0 \wedge f'(0) = 0 \wedge f''$ is a continuous one to one function on some open interval $(a, b): a < 0 < b$ then ...
7
votes
1answer
43 views

A second opinion on a proof in topology

My friend and I were looking over some homework questions for an upcoming test in introductory topology, and one of the questions on the homework was to show that a metric space is normal. What we ...