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

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4
votes
2answers
38 views

Prove by induction that $\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n}=2-\frac{n+2}{2^n}$

Prove by induction that $$ \frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n}=2-\frac{n+2}{2^n} $$ Let $n=r$, so that $$ S_r=2-\frac{r+2}{2^r} $$ Therefore $$\begin{align} ...
0
votes
1answer
43 views

Is this proof about clock hands lining up correct?

Is http://joshuaoldenburg.com/articles/clock-hands-line-up/ a proof? I.e. does it sufficiently prove the times where the clock hands line up? $$ \begin{align} H &= \text{hour (1-12)} \\ M &= ...
3
votes
1answer
65 views

Prove that any subfield of $\mathbb C$ must contain $\mathbb Q$

I just started reading Linear Algebra by Hoffman and Kunze, and I came across the following line: The interested reader should verify that any subfield of $\mathbb C$ must contain every ...
0
votes
1answer
26 views

Adding Sequences proof

Prove that if $a,b : \mathbb{N} \to \mathbb{R} $ are sequences with $\lim_{n \to \infty}{a_n} = L$ and $\lim_{n \to \infty}{b_n} = M$ then $\lim_{n \to \infty}{a_n+b_n} = L + M$ Prove that if ...
1
vote
0answers
34 views

Can someone check my answer to a measure theory question on existence and equality of three integrals.

I have been told to investigate the existence and equality of the integrals; $\int_{[0,1]^2} f\;d\lambda^2$, $\int_0^1\int_0^1 f\;d\lambda(x)d\lambda(y)$ and $\int_0^1\int_0^1 ...
0
votes
1answer
27 views

$f$ is integrable on $[a,b]$ then $\int_{x}^{x+h}\frac{|f(t)-f(x)|}{h}dt\to 0$ if $h\to 0$ for almost all $x$

I am trying to prove if $f$ is integrable on $[a,b]$ then $\int_{x}^{x+h}\frac{|f(t)-f(x)|}{h}dt\to 0$ if $h\to 0$ for almost all $x$ but I am not sure. Attempt - Let ...
2
votes
2answers
36 views

Very easy quiz problem regarding the Archemedean property of R

I had this problem in my quiz on Friday: Consider the following statements: $1$. If $x \in \mathbb{R} $, then there exists some $n \in \mathbb{N}$ with $x < n $. $2.$ If $x,y \in \mathbb{R} $ ...
2
votes
0answers
75 views

Limit of $(\frac{1}{n^2})^{(\frac{1}{n})}$

Question: is this a valid procedure for finding the limit of $$\bigg(\frac{1}{n^2}\bigg)^{(\frac{1}{n})}$$ as $n\rightarrow \infty$. background: This is one of the examples of the failure of the ...
0
votes
1answer
42 views

Prove The Limit Does Not Exist

So I have a few questions in which I have to prove that the given sequence does not have a limit and I'm not too sure if I'm on the right track and if I am what is the next step that I have to do. Can ...
0
votes
0answers
32 views

Prove the limit exists

So I have a couple of problems in which I have to prove that the given limit exists and I'm not too sure if I'm on the right track and if I am what it is that I have to do next. Can anybody give me ...
0
votes
1answer
44 views

Convergent Series 2n-1/2n

Prove the series defined by P(n) = (1 *3 * 5 * (2n-1))/(2*4*6 * (2n)) is convergent It is monotone decreasing and bounded below by zero, but is that enough to say?
1
vote
2answers
60 views

Proving $\lim_{x\to \infty}\ln(x)/x$

Can you please check if my proofs are correct? for $$\lim_{x\to +\infty}\ln(x)=+\infty$$ I used the mean value theorem : $\ln$ continuous on $[1,x]$ $\ln$ differentiable on $(1,x)$ then there ...
0
votes
1answer
14 views

Verification of proof regarding limit and derivative at infinity

Ok so I have been working through Calculus by Spivak and stumbled upon a theorem which I found hard to prove ,and solution in answer book seems to be wrong.So I need you to help me verify my proof. ...
1
vote
2answers
29 views

Symmetric bilinear forms, quadratic forms and matrices

I have computed B=$ \left( \begin{array}{ccc} 0 & 4 & -1 \\ 4 & 2 & 3 \\ -1 & 3 & 1 \end{array} \right) $ Is this correct? If so, even though I may have achieved the correct ...
2
votes
1answer
34 views

If $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ smooth, $ g(x,y)= x^3 + y^3$ and $g \circ f \equiv 0$, then $\det Df \equiv 0$

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a smooth function and $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined by $(x,y) \mapsto x^3 + y^3$. Assume that $g \circ f$ is identically $0$. ...
1
vote
1answer
30 views

A trivial question regarding the supremum of a set

Let $E$ be a subset of nonnegative real numbers. Suppose $E$ is bounded above and put $T = \{ x^2 : x \in E \}$. Say $\alpha = \sup E $. Then, $\alpha^2 = \sup T $. Attempt: Since $E$ is bounded ...
2
votes
0answers
20 views

A different characterization of the infimum of a set

Let $E$ be a set that is bounded below. Let $l$ be a lower bound of $E$. Show that $ l = \inf E $ iff given any $\epsilon > 0$ we can always find $z \in E$ with $z < l + \epsilon $. Attempt ...
1
vote
0answers
35 views

If $u\notin E$, then the supremum of $E\cup\{u\}$ is $\sup\{\sup E, u\}$

Let $S$ be an ordered set and $S \supset E $. Let $\alpha = \sup E \in E$. If $u \notin E$, then we have $ \sup ( E \cup \{u\} ) = \sup\{ \alpha, u \} $. Try: I know this result follows easy by ...
4
votes
3answers
232 views

How to show distributivity in a ring, and what is wrong with my algebra?

I am trying to show the following is a commutative ring with unity, however I am encountering a problem. First, addition and multiplication are defined as: $$a \oplus b=a+b-1$$$$a \odot ...
0
votes
1answer
36 views

Conditions for magic square.

So I've messing around with magic squares and something occured to me: Assume we have a nxn grid of numbers which respects the sum conditions of a magic square as in it has the appropriate column, ...
1
vote
2answers
54 views

Suppose G is a group, p is prime , Then the number of elements of G of order p is multiple of (p-1) [closed]

I need Help . "Suppose $G$ is a group, $p$ is prime , Then the number of elements of G of order $p$ is multiple of $(p-1) $". Give me any advise or note
2
votes
2answers
97 views

There is no homomorphism from $\mathbb{Z_8} \times \mathbb{Z_2} \times \mathbb{Z_2}$ onto $\mathbb{Z_4} \times \mathbb{Z_4}$

If such a homomorphism $\phi$ existed, then the first isomorphism theorem says that $|\ker \phi| = 2$. Since $\mathbb{Z_8} \times \mathbb{Z_2} \times \mathbb{Z_2}$ is abelian, then every subgroup is ...
3
votes
1answer
66 views

If $n\mid m$ prove that the canonical surjection $\pi: \mathbb Z_m \rightarrow \mathbb Z_n$ is also surjective on units

Not sure if this is the right proof (i found it online): Since $n\mid m$, if we factor $m = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, then $n = p_1^{\beta_1}p_2^{\beta_2}\cdots ...
0
votes
1answer
30 views

Lambert W function identity from differential equation

For constants $v,K$ and a function $C(t)$, can you prove that if : $$ \frac{dc}{dt} = - \frac{vc(t)}{K + c(t)},~\text{with } c(0) = c_0 $$ Then the solution: $$ \left[ K \ln c(t) + c(t) ...
0
votes
1answer
12 views

Lambert W function multiplication with scalar

Let $W$ be the Lambert W function, $Y$ be a real valued function and $x \in \mathbb{R} $. Given $ Ye^Y = x \iff Y = W(x) $ is it true that $Y = kW(\frac{1}{k}x)$ for non-zero $k \in \mathbb{R} $ ? ...
0
votes
2answers
44 views

If H and K are finite subgroups of G (another proof )

I have a question and it's solution , but I want another proof if there exist . Thanks
2
votes
1answer
28 views

Trivial mathematical analysis problem

Let $\mathbb{R} \supset E \neq \varnothing$. Put $\alpha = \sup E $. Then, for all $n \in \mathbb{N}$, $\alpha - \frac{1}{n} $ is not an upper bound of $E$, but $\alpha + \frac{1}{n}$ is an upper ...
1
vote
0answers
36 views

Proof of Bolzano-Weierstrass for functions over countable domains

Theorem A bounded sequence of functions defined over a countable domain has a convergent subsequence. The attempted proof uses Bolzano-Weierstrass for real sequences (nested bisections) proof and a ...
1
vote
0answers
21 views

Help with proof about merge two heaps to one heap…

We have two heaps: $H_1,H_2$ that have $n_1,n_2$ elements ($H_1$ have $n_1$ elements and $H_2$ have $n_2$ elements). We know that the smallest element at $H_1$ is bigger the root (the biggest element) ...
1
vote
1answer
27 views

An element $u$ is an upper bound of $E$ if and only if $t>u$ implies $t\notin E$

Let $S$ be an ordered field and $S \supset E\neq \varnothing$. Then, the following are equivalent: $u \in S$ is an upper bound of $E$. $t \in S$ and $t > u$ implies $t \notin E $. My Try: ...
0
votes
0answers
14 views

Surjective $\gamma \colon I \to M^1$, $\gamma (t_1)=\gamma (t_2)$ can be extended to a periodic parametrization of $M^1$

Suppose that $\gamma \colon I \to M^1$ is a smooth surjective curve in a Riemannian connected 1-dimensional manifold. Furthermore, suppose that it is parametrized via arc lenght i.e.:$$||\dot ...
3
votes
0answers
25 views

On the greatest lower bound property

Proposition: Let $S$ be an ordered field and $S \supset E \neq \varnothing $. $E$ is bounded below. Then $ \inf E = - \sup ( - E ) $ Try: Write $- E = \{ -x : x \in E \} $ and let $l $ be a lower ...
0
votes
1answer
52 views

Use Resolution to proove a sentence in First Order Logic

I was just wondering if anyone could tell me if I've solved this problem right. If wrong, I would like to know what I did wrong. "Use resolution to prove Green(Linn) given the information below. You ...
0
votes
1answer
19 views

Another characterization of the supremum of a set

$u$ is an upper bound of a set $E \subset S$ if given any $\epsilon >0$, there is $\delta \in E $ such that $u - \epsilon < \delta$. PROBLEM: An upper bound $u$ of $E \subset S$ ($E \neq ...
0
votes
1answer
60 views

System with two quadratic equations

Respected All. I am unable to find out what's so wrong in the following. Please help me. It is given that $t$ is a common root of the following two equations given by \begin{align} &x^2-bx+d=0 ...
0
votes
4answers
55 views

Is this a valid proof of the Quotient rule?

In an emergency in high-school, I once derived the quotient rule from the chain and product rules. I now wonder whether this was actually a valid proof. I reconstructed it as well as I could remember: ...
1
vote
2answers
53 views

Prove there exists dense open set

Let $G$ be an open set in $X$ and $D$ be a dense open set in $G$.Show there exists a dense open subset $V$ of $X$ such that $V\cap G=D$. Since $D$ is open in $G$, there exists $V$ open in $X$ ...
1
vote
1answer
31 views

big-Oh prove or disprove 2^n is in big-Oh(3^n)

the definiton of Big-Oh says $\exists c\in$R+,$\exists B\in$ N,$\forall n\in$N, $n \geq B$$\implies$$2^n \leq c\times 3^n$. I believe $2^n \in O(3^n)$, but how to prove it? can anyone help. This this ...
0
votes
0answers
22 views

Do these statements prove this formula?

$$ \frac{d^n}{dx^n}[g(x)^{f(x)}] = g(x)^{f(x)} B_n(d_1,\cdots,d_n) $$ Calling $$ d_n = \frac{d^n}{dx^n}[ln(g(x))f(x)] $$ Since faa di bruno's formula states $$ \frac{d^n}{dx^n}[f(g(x))] = ...
1
vote
0answers
36 views

Proof Verification for $n2^{n-1} = \sum\limits_{k=1}^n k\binom{n}{k}$

$(1+x)^n = \sum\limits_{k=0}^n\binom{n}{k}x^k$ by binomial theorem $\frac{d}{dx}(1+x)^n =\frac{d}{dx}\sum\limits_{k=0}^n\binom{n}{k}x^k$ $n(1+x)^{n-1} = \sum\limits_{k=1}^n k\binom{n}{k}x^{k-1}$ ...
1
vote
1answer
24 views

Give an combinatorial argument

I need to find the possible value of $R_i$ and prove it by giving combinatorial argument, for following identity. I was able to give an argument like this. Consider double counting. Count ...
2
votes
1answer
40 views

Direct sum of simple modules and Schur's Lemma

Suppose $M,N$ are two non-isomorphic simple $R$-modules. For $m,n\geq1$, is it true that $$ \text{Hom}_R(M^{\oplus m},N^{\oplus n})\cong\hat{0}\,? $$ I think it's true by Schur's Lemma. ...
0
votes
1answer
58 views

King Arthur and knights at the round table puzzle

Can you help me with this math problem: Each of the K knights from the round table needs to choose a card which is marked with a number from 1 to N, N >= K. The cards all have different number. ...
2
votes
2answers
76 views

Show that a Series Diverges

Question: Let a sequence ($a_n$) have the property $\lim \limits_{n \to \infty} na_n = a > 0$ Show that the series $\sum_{n=1}^\infty a_n$ diverges Attempts: Basically, I firstly tried ...
1
vote
2answers
59 views

Is my proof regarding continuity at irrationals correct?

Consider the Thomae's function $$f(x)=\begin{cases} 0 \text{ ; when } x \text{ is irrational} \\\frac 1 q \text{ ; for } x=\frac p q \text{ irreducible fraction}\end{cases}$$ In the following proof ...
5
votes
1answer
39 views

Sequence of $f_n \in R[0, 1]$ that converges pointwise to $f \in R[0, 1]$ such that $\lim_{n \to \infty} \int_0^1 f_n dx \neq \int_0^1 f dx$.

Question: Find a sequence of functions $f_n \in R[0, 1]$ that converges pointwise to $f \in R[0, 1]$ such that $\lim_{n \to \infty} \int_0^1 f_n dx \neq \int_0^1 f dx$. ($R$ means ...
2
votes
2answers
46 views

Prove if $3$ does not divide $n$, then $n^2=1+3k$ for some integer $k$

I am proving by cases but am getting confused. I am not sure if this leads to a contradiction or not. Here's what I have so far: Direct Proof. Suppose $3$ does not divide $n$. Case 1: remainder ...
12
votes
5answers
259 views

How to prove $n < \left(1+\frac{1}{\sqrt{n}}\right)^n$

I want to know how to prove the following inequality. For $n = 1, 2, 3, \ldots $ $$ n < \left(1+\frac{1}{\sqrt{n}} \right)^n $$ I tried with math induction but I failed.
4
votes
0answers
29 views

Every $\sigma-$finite measure is semifinite. $(X, \mathcal{M}, \mu)$ is a measure space.

Definition 1: Say $X = \bigcup_{n=1}^{\infty} E_n $ where $E_n \in \mathcal{M}$ and $\mu( E_n ) < \infty $ for all $n$, we call $\mu$ $\sigma$-finite. More generally, if $E = \bigcup^{\infty} E_n ...
2
votes
1answer
61 views

Find the chance that $a^3 + b^3 \equiv 0 (\mod 3)$

We are given set of integer numbers $\{1,2, \dots N\}$. $N \ge 3$ Then perform a drawing with replacement of two elements $a$ and $b$. Problem is to find the probability of following statement holding ...