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

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6
votes
6answers
168 views

Show that $\lim_{n\to\infty}\frac{a^n}{n!}=0$ and that $\sqrt[n]{n!}$ diverges.

Let $a\in\mathbb{R}$. Show that $$ \lim_{n\to\infty}\frac{a^n}{n!}=0. $$ Then use this result to prove that $(b_n)_{n\in\mathbb{N}}$ with $$ b_n:=\sqrt[n]{n!} $$ diverges. ...
1
vote
1answer
27 views

The annulus $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open

I want to prove that the set $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open. This is my attempt. Let $z \in A_{r,s}(z_0)$. Then $|z-z_0|-r>0$. Let $r'=[|z-z_0|-r]/2$. Then ...
0
votes
1answer
55 views

Number of Lattice Points in a Triangle

Problem Let the co-ordinates of the vertices of the $\triangle OAB$ be $O(1,1)$, $A(\frac{a+1}{2},1)$ and $B(\frac{a+1}{2},\frac{b+1}{2})$ where $a$ and $b$ are mutually prime odd integers, ...
1
vote
3answers
255 views

Can the proof of Theorem 1.20 (b) in the book, The Principles of Mathematical Analysis by Walter Rudin, 3rd ed., be improved?

I'm reading Walter Rudin's PRINCIPLES OF MATHEMATICAL ANALYSIS, third edition, and am at Theorem 1.20(b), where he states and proves that between any to real numbers, there is a rational; that is, if ...
1
vote
1answer
90 views

Relation between the covers by sets of small diameter and the size of uniformly separated sets

Sorry I didn't find a better title. Here is the problem and my solution so far, I'd appreciate if someone could told me if is correct and for the last point, which at first sight seems to be ...
0
votes
0answers
28 views

Proving a property of the largest limit point

Let $(a_n)_{n\in\mathbb{N}}$ be a bounded sequence. By Bolzano-Weierstraß this sequence does have a limit point. Let $\bar{a}$ denote the largest limit point of the sequence. Show that among ...
0
votes
2answers
66 views

Proving formally $\lim_{x \to -\infty}\mathrm{Pr}( \left \lfloor{x}\right \rfloor \le X < x) = 0$ (Proof check)

we have $$\lim_{x \to -\infty}\mathrm{Pr}( \left \lfloor{x}\right \rfloor \le X < x) $$ where X is a real random variable, and $x \in R$. My idea of a proof would be by contradiction: Assume ...
2
votes
1answer
27 views

Proving $\{x_n\}$ converges to $a$ when $|x_n-a|\le Cb_n$ for large $n$ and $C$ is a positive constant.

If $\{b_n\}$ is a sequence of nonnegative numbers that converges to $0$, and $\{x_n\}$ is a real sequence that satisfies $|x_n-a|\le Cb_n$ for large $n$, where $C$ is a fixed positive constant, prove ...
2
votes
2answers
167 views

Help understanding proof of Theorem 2.43 in Baby Rudin

Theorem $\hspace{5 pt}$ Let $P$ be a nonempty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable. Proof $\hspace{5 pt}$ Since $P$ has limit points, $P$ must be infinite. Suppose $P$ is ...
0
votes
0answers
137 views

Prove $x_n$ converges to $a$ iff every subsequence of $x_n$ also converges to $a$.

Prove $x_n$ converges to $a$ iff every subsequence of $x_n$ also converges to $a$. Suppose that $\{x_n\}$ is a sequence in $\mathbb R$. Definitions available: (1) A sequence of real numbers ...
1
vote
2answers
80 views

Is it true that if $x_n$ converges and $y_n$ is bounded, then $x_ny_n$ converges?

Is it true that if $x_n$ converges and $y_n$ is bounded, then $x_ny_n$ converges? $x_n$ is said to be bounded if and only if it is bounded both above and below. I believe this to be false. My ...
2
votes
1answer
43 views

Powerset of $A\times B$

$A = \{0,1\}$ and $B = \{1,2\}$, find $P(A\times B).$ And I found $A\times B = \{(0,1),(0,2),(1,1),(1,2)\}$ So if I wanted $P(A\times B),\text{would I do this:} \\ P(A\times B) = ...
2
votes
1answer
49 views

Field Theory : What is wrong with this “homomorphism”?

Let $E/F$ be a field extension and $\alpha \in E$ be algebraic over $F$. Let $m(x) \in F[x]$ be irreducible and such that $m(\alpha) \neq 0$. Define $$ \phi : F(\alpha) \to F[x]/(m) $$ by ...
2
votes
1answer
177 views

Attempt to prove that every real number is a limit of a sequence of rational numbers

Prove that given a real number $x$, there exists a rational sequence $r_n$ such that $r_n \to x$ as $n$ grows. Proof: Suppose $x$ is a real number. Then we know by definition, there exists a ...
0
votes
1answer
955 views

Proving graph connectedness given the minimum degree of all vertices

I know that this is a repeat of a previous question asked with a similar title, but I didn't want to revive an old thread. The solution presented in that thread seems to be the common one, but I was ...
2
votes
1answer
227 views

Every compact set in $\mathbb{Q}$ has empty interior proof

Consider $\mathbb Q$ with the subspace topology. I read on planet math website that every compact subset in this space has empty interior and then I tried to prove it. Please could someone tell me if ...
1
vote
2answers
47 views

Showing the summation of numbers

Using each of the digits 1 through 9 once, form numbers whose sum is 100. If you think it can't be done, then prove it. My attempt: I say it can't be done because the sum of all numbers $1-9$ is ...
0
votes
0answers
35 views

Lebesgue measure of decreasing sets. Something wrong with this proof?

$\mathcal{M}$ denotes the collection of Lebesgue-measurable sets and $\lambda$ the Lebesgue measure. I have the following exercise: Suppose $\lbrace E_n \rbrace^{\infty}_{n=1} \subset ...
3
votes
1answer
162 views

Question about proof of Browder, Minty Theorem

Could someone please assist with the following question: Refer to the following post.
1
vote
2answers
71 views

Complex Analysis problem and solution

Suppose $f(z)$ is holomorphic on $D(0,1)$ such that $|f(z)| \leq 1, \forall z \in D(0,1)$ and $f$ has zero of order $3$ at $z=0.$ Prove that $|f^{\prime\prime\prime}(0)| \leq 6$ and determine the ...
1
vote
1answer
61 views

Show an operator is compact if $\sum \|Te_n\| < \infty$

Let $H$ be a separable Hilbert space, define a bounded linear operator $T:H \rightarrow H$, show it is compact if $\sum \|Te_n\|_H < \infty$. My attempt: We show that $T(B)$ is totally ...
2
votes
2answers
83 views

Proving Equality About The Null Space

let there be a matrix $A^{n*m}$ that $Ax=b$ the solution set of the homogeneous system $H=(h\in F^m; Ah=0)$ the solution set of the non-homogeneous system $L=(l \in F^m; Al=b)$ Prove: if $l_0\in ...
2
votes
1answer
77 views

Proving $\sum_{k=1}^{\infty}\frac{\sin kx}{x}=\frac{\pi-x}{2}$ for $0\le x\le 2\pi$

Refer to this OP: Sign of a series, we have the following equation \begin{equation} \sum_{k=1}^{\infty}\frac{\sin kx}{k}=\frac{\pi-x}{2} \end{equation} defined for $0\le x\le 2\pi$. Here is ...
2
votes
3answers
52 views

How do I prove that the inductive sequence $y_{n+1}= \dfrac {2y_n + 3}{4}$ is bounded . $y(1)=1$

How do I prove that the inductive sequence $y_{n+1}= \dfrac {2y_n + 3}{4}$ is bounded? $y(1)=1$ Attempt: Let us assume that the given sequence is unbounded. : Then, $y_{n+1} \rightarrow \infty$ ...
2
votes
2answers
45 views

Verify my proof: Let $ f $ and $ g $ be functions. Show that, if $ g(f(x)) $ is injective and $ f $ is surjective, then $ g $ is injective.

Could someone verify my proof? Proposition: Let $ f: X \rightarrow Y $ and $ g: Y \rightarrow Z $ be functions. Show that, if $ g(f(x)) $ is injective and $ f $ is surjective, then $ g $ is ...
1
vote
0answers
41 views

Given $A_n : X\rightarrow Y$ linear and continuous, for each $x\in X$ $A_n(x) \rightharpoonup A(x)$ in $Y$, is $A$ continuous?

Given $A_n : X\rightarrow B$, a linear and continuous operator between two Banach spaces, for each $x\in X$ $A_n(x) \rightharpoonup A(x)$ in $Y$, is $A$ linear and continuous? My attempt: $A$ ...
1
vote
1answer
97 views

Exercise about an operator (adjoint and spectrum)

Let $y\in c_0$ and define the operator from $l^2 \rightarrow l^2$ as the following $$T\bigg(\sum x_n e_n\bigg) \mapsto \sum y_n x_n e_n.$$ I have shown that the operator is continuous, compact and ...
3
votes
1answer
106 views

Is this proof that $\lfloor x \rfloor \geq n \left\lfloor \frac{x}{n} \right\rfloor$ correct?

In this text the fractional part of a real $x$ shall be denoted $\{x\}$, such that $x = \lfloor x \rfloor + \{x\}$. Theorem: $$ \forall x \in \mathbb{R}_{\geq 0} \forall n \in \mathbb{N}_{\geq 1} : ...
3
votes
1answer
81 views

How to prove this version of the Cantor-Schroder-Bernstein theorem?

My text states the Cantor-Schroder-Bernstein theorem as follows: Suppose that $X$ and $Y$ are non-empty sets such that $|X|>|Y|$. Then, any function $f:X\rightarrow Y$ is not an injection, i.e. ...
0
votes
0answers
40 views

Finite dimensional spaces and R^n

I have a couple of questions, any assistance would be appreciated. I know that it can be shown that any finite dimensional space $M$ of dimension $N < \infty$ endowed with an inner product can be ...
25
votes
3answers
2k views

Can't find mistake in an easy proof.

Consider the following theorem. $\textbf{Theorem:}$ for any sets $A, B, C, D$, if $A \times B \subseteq C \times D$ then $A \subseteq C$ and $B \subseteq D$. Then the following proof is given. ...
0
votes
0answers
51 views

Change of measure

Title might be wrong. Anyway I am given that $\delta$ is nonnegative and measurable in $(\Omega, \mathfrak{F}, \mu)$. It can be shown that the mapping $\nu: \mathfrak{F} \to [0, \infty]$ defined by ...
0
votes
0answers
81 views

Cumulative distribution function for a Poisson distribution

This is a past exam question and I just want your opinions on if it's sufficient or not. I had to prove: Let the discrete random variable $X$ have a Poisson distribution with parameter $\lambda$. ...
3
votes
3answers
81 views

Prove or Disprove Inequality By Induction

Prove or Disprove $\sum_{i=0}^n(2i)^3 \le (8n)^3 $ If true, prove using induction. If false, give the smallest value of n that is a counter example and the values for the left and right hand sides ...
0
votes
0answers
67 views

Using jugs filled with water problem

Given jugs $m$ and $n$ liters (WLOG $m<n$) is it always possible to get all $i$, $0 \leq i \leq n ?$ If so, prove it. If not, explain which $i$ you can get. Is there also a minimum number ...
0
votes
1answer
42 views

Correctness of proof that the commutative operation * on a binary structure is a structural property [duplicate]

Here is what I have as a proof for now. Can you tell me where I need to edit it and or how it should be instead? Let and be two arbitrary binary structures with an isomorphism f: S->T. Assume * is ...
1
vote
1answer
108 views

Finding bases for subspaces of $F^5$

Find bases for the following subspaces of $F^5$: $W_1 = \{(a_1,a_2,a_3,a_4,a_5) \in F^5 : a_1-a_3-a_4=0\}$ $W_2=\{(a_1,a_2,a_3,a_4,a_5) \in F^5 : a_2=a_3=a_4, a_1+a_5=0\}$ What are the dimensions of ...
1
vote
0answers
31 views

Validity of Induction Proof - $\{ \land, \top, \bot \}$ is an incomplete set of connectives

I need to verify a proof of the fact that $\{ \land, \top, \bot \}$ is not complete. I consider $\top$ and $\bot$ to be $0$-ary logical connectives that are constantly true or false. That is ...
2
votes
2answers
86 views

$\phi_n \rightarrow \phi$ weakly-$*$, then $\|\phi\|\leq \limsup_n \|\phi_n\|$.

$\phi_n \rightarrow \phi$ weakly-$*$, then $\|\phi\|\leq \limsup_n \|\phi_n\|$. My attempt: $$\|\phi\| = \sup_{\|x\| = 1} |\phi(x)| \leq \sup_{\|x\| = 1} \lim_n |\phi_n(x)|$$ Using an ...
1
vote
2answers
164 views

retraction induced homomorphism is surjective

Im having a hard time proving this although it looks trivial... Let $r:X\to A$ be a retraction between a topological space $X$ and $A\subset X$ such that $r(a_0)=a_0$ for $a_0\in A$ then the induced ...
0
votes
1answer
25 views

Sequence of irrational numbers

I have to show that: If $x$ is an real number, there is a sequence of irrational numbers converging to $x$. My attempt: We know that every $x$ real is an accumulation point of the irrational ...
0
votes
1answer
49 views

Prove that a sequence converge.

I need to do this exercise: Assume $0 \le a \le b$.Do the sequence $\{(a^{n} +b^{n})^{1/n}\}$ diverge or converge?. If the sequence converge find the limit. Well what I did is: I computed the limit ...
1
vote
2answers
459 views

Prove that if all edge-costs are different, then there is only one cheapest tree.

Prove that if all edge-costs are different, then there is only one cheapest tree (minimum spanning tree). (Use contradiction and make sure to keep track of the costs of the different trees involved.) ...
2
votes
0answers
65 views

Is it true that $\operatorname{Int}(A) \cap \operatorname{Int}(B) = \operatorname{Int}(A \cap B)$?

can someone please verify my proof? (a) Is it true that $\operatorname{Int}(A) \cap \operatorname{Int}(B) = \operatorname{Int}(A \cap B)$? (b) Is it true that $\bigcap ...
0
votes
0answers
83 views

Subsequences of a sequence converging and the Bolzano Weierstrass theorem

I need to prove the following: Let $\{a_{n}\}$ be a bounded sequence of real numbers.Prove that $\{a_{n}\}$ has a convergent subsequence.(Hint: You may want to use the Bolzano-Weierstrass Theorem) ...
0
votes
2answers
45 views

If $U$ is open, is it true that $U = \operatorname{Int}(\overline{U})$?

Can someone please verify this proof? I am aware that there must be a similar question elsewhere, but I need help with my proof in particular. If $U$ is open, is it true that $U = ...
0
votes
1answer
34 views

Prove the convergence of a sequence.

Prove that $$\left\{\frac{{n+k \choose k}}{(n+k)^k} \right\}_{n=1}^\infty \longrightarrow \frac 1{k!}$$ where $${n+k \choose k}=\frac{(n+k)!}{n!k!}.$$ My attempt for the question We only have to ...
1
vote
1answer
880 views

Prove that if a sequence $\{a_{n}\}$ converges then $\{\sqrt a_{n}\}$ converges to the square root of the limit.

My attempt and the question:Can you tell if I am right :)? thank you
5
votes
2answers
51 views

Show that $\bigcup \overline{A_\alpha} \subseteq \overline{\bigcup A_\alpha}$

Can someone please verify my proof? I am aware that there is a similar question elsewhere, but I need help with my proof in particular. Show that $$\bigcup \overline{A_\alpha} \subseteq ...
0
votes
0answers
26 views

Problem with measure theory argument.

Could someone point out the flaw in this reasoning concerning the equality of outer and inner measure on sets with finite measure? Let $\epsilon >0$, and let $S\subset \mathbb{R}^n$ with $m^*(S) ...