**17.676 proof-verification questions.**

Let $G$ be a complete bipartite graph of $7$ vertices. $G$ is also planar and has Eulerian path. What is $G$'s chromatic number?
I think that because $G$ has Eulerian path then it must have two ...

I try to solve using laplace transform
$y'' + y = \sin (t)$ with $y(0) = 1$ and $y'(0) = 2$
but I don't get a solution, I don't know why. I check my work and it seems fine.
My calculations
Is ...

My friend's assignment has the following question:
Let $F$ be a field and $E=F(\alpha)$, where $\alpha\notin F$ and $\alpha^2 \in F$. Find the set of all $\beta \in E$ such that $\beta^2 \in F$.
...

Cantor set
See the link, I am referring to cantor set on real line. I wish to show that it is compact. I am doing this buy pointing following arguments. I am not sure if this is enough.
Cantor set ...

A function $f$ is called convex on an interval $[a, b]$ if, for any $x, y \in
[a, b]$ and $t \in [0, 1]$ we have $f(tx + (1 − t)y) \leq tf(x) + (1 − t)f(y)$.
Would drawing a picture of this help in ...

$(a,b)$ denotes G.C.D of $a$ and $b$
Let $(b,c)=d$. So, $d|b$ and $d|c$.
Claim : $(ab,ac)=ad$. As d|b and d|c , so $d|b.b...b$ ($a$ times) which means $d|ab$. similarly $d|ac$. So d is common ...

I have no stats training, so I am asking if I am attacking this simple statistical problem correctly.
What are the chances of flopping a royal flush in Texas hold’em?
My attempt: The first card ...

Let $X\subset Y\subset Z$. Prove that $Z-(Y-X)=X\cup(Z-Y)$.
Here, $A-B$ is the complement of $B$ in $A$.
To establish equality, I need to show that $Z-(Y-X)\subset X\cup(Z-Y)$ and $X\cup(Z-Y)\subset ...

I am really not sure of my answer, could someone check it for me? Thank you
...

let $\mathbb{C}^*$ be a multiplicative group of non-zero complex numbers.
Suppose that $H$ be a subgroup of finite index of $\mathbb{C}^*$.Prove that $H=\mathbb{C}^*$
Attempt
If $[\mathbb{C}^*.H]=n$...

Consider a particle $X_t^1$ which moves in space and assume that we have a countable family of exponential times ($exp(\nu)$ distributed) $T_1,....,T_N$ $N \in \mathbb{N}$, where $T_1$ is the time ...

I was asked for a homework to show that the pair $(I^n, \partial I^n)$ is homotopy equivalent to $(R^n, R^n\setminus \left\{0\right\})$, where a relative homotopy between two maps $f, g: (X, A)\...

So my friend gives me a problem.
Solve for $x$ in the following equation: $$x^3 - 8 = 0$$
So I did the following: $$\begin{align} x^3 \require{cancel}\cancel{- 8} \cancel{+ 8} &= 0 + 8 = 8 \\ ...

Let $K\subseteq\mathbb{C}$ be compact and non-empty. Show there exists a $z\in K$ such that $|z|=\inf\{|w|:w\in K\}$.
Is such a $z$ in general unique? Give an example of a non-compact $K$ such that ...

For each and every $k\geqslant 1$, I want to prove whether or not the following equation has infinitely many solutions $(a_1,a_2,b,x)$ over the integers, and on the condition that $a_1\neq a_2\neq b$.
...

Sketch $M=\{re^{i\phi}:r\in(1,2],\phi\in[0,\pi)\}$ and give its closure $\overline{M}$. Is $M$ compact?
$\overline{M} = \{re^{i\phi}:r \in (1,2], \phi \in [0,\pi) \} \cup \{ re^{i\pi} \} \cup \{e^{i\...

The questions that have a blue section (Preformatted text), I am not sure why the proof is correct and need some explanation. The other problems I am not sure if I had done them correctly.
(4) Show ...

The questions that have a blue section (Preformatted text), I am not sure why the proof is correct and need some explanation. The other problems I am not sure if I had done them correctly.
(1) ...

I'm working through the following problem
Let $1 < p < \infty$ and $f\in L^p([a,b])$. Now, if $F(x) = \int_a^x f(t)\,dt$, show that there exists $K \in \mathbb{R}$ such that for every ...

I would just like to confirm that I'm doing this correctly. If not, any help would be appreciated. Thanks!
The problem:
A painting sold for $\$274$ in $1977$ and was sold again in $1987$ for $\$470$....

Show $f:[1,\infty[\rightarrow[1,\infty[, x\mapsto x^x$ is bijective and continuous, and show continuity for its inverse function $f^{-1}$.
My idea here is to proceed with (hopefully) minimal effort ...

Say $f_n:[0,1]\to [0,1]^d$ is the $n$-th iteration of a $d$-dimensional Hilbert curve touring its range. Is it true that for any open $S\subset [0,1]^d$, then amount of time $f_n$ spends in $S$ is ...

Let ~ be a equivalence relation on $\mathbb{R}$ such that $\mathbb{R}/$~ is finite.
I need to show that tue quotient space $\mathbb{R}/$~ can't be a Hausdorff space.
My attempt: Supposse that $X=\...

Find the limit of $d_1 = 1, d_{n+1}=1+\frac{d_n}{3}$
We show $d_n\leq(3/2)$ by induction.
$d_1=1 = (2/2)\leq(3/2)$ and so our hypothesis is that $d_n\leq(3/2)$.
$n+1:$
$d_{n+1} \leq 1 + \frac{(3/2)...

I am trying to show the following,
Let $f : \mathbb{R^+} \to \mathbb{R}$ be a real-valued function such that $$\lim_{x \to \infty} f(x) = \lim_{x \to 0} f(x) = 0$$
Let $g : \mathbb{R^+} \to \...

I am working on an assignment in discrete structures and I am blocked trying to prove that $\sqrt{11}-1$ is an irrational number using proof by contradiction and prime factorization.
I am perfectly ...

From Friedberg....
" Let $V=\{(a_1 ,a_2):a_1 ,a_2\in\mathbb F\}$, where $\mathbb F$ is a field. Define addition of elements of $V$ coordinatewise, and for $c \in \mathbb F$ and $(a_1 , a_2) \in V$, ...

The question seems trivial which is why I have some trouble coming up with a proof that is mathematically correct. BTW I cannot yet use eigenvalues as we have not yet covered them in class.
If
$$D=[...

I'm studying Multivariate Calculus and I've just studied the inverse theorem and now I'm doing some exercises.
There are some questions about local injectivity that are causing some doubts. For ...

Let $f: [a,b]\rightarrow\mathbb{R}$, $x\mapsto x^2$ where $a,b\in\mathbb{R}$
Without loss of generality, assume $|a|>|b|$
Let $\epsilon>0, x$ and $y\in[a,b]$ and $\delta=\min\{1,\frac{\epsilon}...

I'm trying to understand the use of the Compactness Theorem to proof certain properties for theories in languages. I've tried to prove the following:
If $\phi$ holds in every model of the theory $T$, ...

Here is Prob. 5, Sec. 24, in the book Toplogy by James R. Munkres, 2nd edition:
Consider the following sets in the dictionary order. Which are linear continuua?
(a) $\mathbb{Z}_+ \times [0, ...

Test $$\sum_{k=1}^{\infty}\left(\frac{k+1}{k}\right)^{k^2}3^{-k}$$ for convergence and absolute convergence.
We apply the ratio test for $\displaystyle \sum_{k=1}^{\infty}\left|\left(\frac{k+1}{k}\...

Suppose that $A$ is a commutative ring with $1$ and suppose that $\forall x \in A \exists n >1 \in \mathbb{N}$ dependent from x such that $x^n=x$. Prove that if $I$ is a prime ideal then $I$ is ...

I have this question of the Hatcher book:
Show that if a path-connected, locally path-connected space $X$ has $\pi_1(X)$ finite, then every map $X\rightarrow S^1$ is nullhomotopic. [Use the ...

I think I've proven this homework problem, but I'm not sure if my proof is correct:
Given an integer $n$, prove that there exists at least one $k$ for which $n\mid \phi(k)$. (Where $\phi$ is the ...

Suppose that $f$ is differentiable on $[0,1]$, $f(1)>f(0)$ and $f'(0)=f'(1)=0$. Show that for every $y\in[0,f(1)-f(0)]$ there exists $x\in[0,1]$ such that $f'(x)=y$.
Since $f$ is differentiable on ...

The question is as follows:
Let f be a monomorphism from a lattice $L$ to a lattice $M$.Show that $L$ is isomorphic to a sublattice of $M$.
My attempt:
Since $f$ is a monomorphism from a lattice $L$...

Please check if my proof contains any error! Thank you so much!
Lemma: Let $A=\{J\subseteq\{1,2,\cdots,k\}\mid J\neq\emptyset\}$ and $B=\{J\subseteq\{1,2,\cdots,k+1\}\mid J\neq\emptyset\}$, then $$B-...

Assume it is rational. So $6 = \frac{p^2}{q^2}$ and $(p,q)=1$. So $p^2 = 6q^2 = 2 (3q^2)$. So $p^2$ is even and so $p$ is even. Let $p=2r$. So $p^2 = 4r^2$. Putting back in the equation, I get $2r^2=...

I wrote up the following "proof". I've marked bits that feels especially iffy. Naturally the solutions had a much better proof than mine :) But I would like to know how well mine holds up to scrutiny.
...

I am tasked with showing that
If $a,b\in \mathbb{R}$, show that $\max{\{a,b\}}=\frac1{2}(a+b+|a-b|)$
I think I can say "without loss of generality, let $a<b$." Then $b-a>0$ But also, $$\...

Is the following argument Correct?
- Let $G$ be a group such that $g^2=1$ for all $g\in G$. Prove that $G$ is abelian.
Proof. Let $g\in G$ from the axiom of inverse we know that there exists a $g^{-...

Test $x_1=0, x_{n+1}=\frac{9+(x_n)^2}{6}$ for convergence and find its limit.
Note that $x_1=0, x_2=\frac{9+0²}{6}=\frac{9}{6}, x_3=\frac{9+9/6}{6}=\frac{17}{9}\dots$ with $x_1 < x_2 < x_3$. $(*...

First, the equality holds, as:
$$\lim_{n\to\infty}\sum_{k=0}^n\binom n k \left(\frac x n\right)^k =\lim_{n\to\infty}\left(\left(1+\frac{x}{n}\right)^n\right) = e^x =
\sum_{k=0}^\infty \frac{x^k}{k!} ...

Let $f$ be a $1$-periodic and continuous function on $\mathbb{R}$. I would like someone to verify that the following proof is correct. I prove that $\|\sigma_n(f)-f\|_\infty\to0$, where $\sigma_n(f)$ ...

I'm trying to make this proof short and compact, so suggestions to make the proof crisper are welcome.
I will show that a set of incongruent integers is a complete residue system.
Let $\Lambda$ ...

I wanted to check that the proof I wrote for the following question is correct.
Let $H = [a,b] \times [c,d] \subset \mathbb{R}^2$, and suppose $f: H \rightarrow \mathbb{R}$ is continuous and $g:[a,...

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