0
votes
0answers
19 views

Correctness of Proof that the limit of $\sqrt n\cdot c^n$ as $n$ tends to $\infty$ is $0$

The problem is: Given that $|c|<1$ prove that $\lim_{n\to\infty} \sqrt n \cdot c^n =0$. I am asked to use the comparison lemma and archimedean property to show convergence for sequence $\{1/\sqrt ...
0
votes
0answers
20 views

Correctness of Proof that the Archimedean Property of Reals is equivalent to lim $1/n$ as n tends to infinity

Here's what I have gotten so far: The Archimedean Property states 1) For every $\epsilon$ >0 there is a positive integer n s.t. $1/n$< $\epsilon$ and 2) For every positive number c there is a ...
1
vote
1answer
30 views

Where was I supposed to use compactness in this proof that a compact subspace can be separated from a point by open sets?

Can anyone check my proof below? P. Let $X$ be a metric space. Prove that if $K\subseteq X$ is compact and $x\notin K$, there exist disjoint open sets $U$ and $V$ such that $K\subseteq U$ and ...
3
votes
1answer
27 views

Every compact set $S\in \mathbb{C}$ is bounded

This is my proof for every compact set $S \subseteq \mathbb{C}$ is bounded. Let $S \subseteq \mathbb{C}$ be compact and assume that it is not bounded. Then for each $z\in \mathbb{C}$ and for each ...
0
votes
0answers
32 views

Correctness of proof based on nested interval theorem

I am given that for positive integer n, let $a_n=1-(1/n)$ and $b_n=1+(1/n)$. $I_n=[a_n,b_n]$. I am to show that 1) the hypothesis of the nested interval theorem are satisfied, 2) find the point of ...
1
vote
1answer
61 views

Proof that arithmetic and geometric mean converge

I need some help with understanding a part of this proof and also writing it up correctly. Given $a_n\geq a_{n+1}\geq b_{n+1} \geq b_n$ with $a_1=a$ and $b_1=b$. I am also given that ...
0
votes
1answer
38 views

$D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact

This is the proof I wrote for $D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact. $$ \bigcup_{n=2}^{\infty}D_{1-(1/n)}(0) $$ is clearly a open covering of $D_1(0)$. Consider the finite ...
0
votes
1answer
38 views

Prove that a function has limit everywhere.

I need to prove the following: Assume that $f: \mathbb{R} \to \mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ for all $x,y \in \mathbb{R}$. If $f$ has a limit at zero, prove that $f$ has a limit at every ...
6
votes
6answers
113 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
2answers
82 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 ...
0
votes
0answers
19 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 ...
1
vote
2answers
87 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
1answer
21 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
29 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 ...
0
votes
0answers
29 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
1answer
33 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
36 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
1
vote
2answers
108 views

An example of set with a countably infinite set of accumulation points

I have to give An example of set with a countably infinite set of accumulation points, and I say: We can consider the set or real numbers and we take an arbitrary real number $x$ then the interval ...
0
votes
2answers
18 views

Convergence of a sequence in absolut value.

I need to prove this: If $a_{n}$ converges to $A$, then $|a_{n}|$ converges to $|A|$. And I have this: $a_{n} \rightarrow A$ then, given $\epsilon>0$ there exists $N \in J$ such that ...
0
votes
1answer
105 views

Few Questions about analysis in Rudins book

I have been looking at intro to real analysis. I am using the text book "Principals of Mathematical Analysis, third edition" by Walter Rudin. I have some questions about things I found confusing and ...
3
votes
3answers
463 views

Is this proof of the fundamental theorem of calculus correct?

A student friend of mine recently gave me a proof of the fundamental theorem of calculus which does not correspond to any I can find in the textbooks. It starts by considering an increasing continuous ...
1
vote
1answer
43 views

Constructing a specific normalised test function.

Given the following class of normalised test functions: \begin{equation} \mathcal{T}=\{\phi\in C_c^{\infty}(\mathbb{R}^n)\ |\ \text{supp }\phi \subseteq B(0, 1),\ \|D\phi\|_{\infty}\leq 1\} ...
3
votes
2answers
79 views

Topology of test functions $\mathcal{D}(\mathbb R)$

(My motivation for the following question is to understand the distribution theory) The space of test functions: $\mathcal{D}(\mathbb R)= \{\phi:\mathbb R \to \mathbb R : \phi \in C^{\infty}(\mathbb ...
4
votes
2answers
77 views

$f'$ is bounded and isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\lim_{x\to y}f'$ does not exist

Prove/disprove: $f$ has a bounded derivative and $f'$ isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\displaystyle\lim_{x\to y}f'$ does not exist. I think that if $f'$ ...
0
votes
1answer
46 views

Rudin: A compact metric space $K$ has a countable base, therefore $K$ is separable.

Hi this is a problem from Rudin's Princ. of Mathematics. I was hoping someone could check this part of my proof for the following question, comments would be very appreciated!: $25.$ Prove that ...
0
votes
1answer
33 views

Proof about limit of derivative (Including Answer)

If the limit of $f(x)$ exists and is finite, and the limit of $f'(x)$ exsits and is equal to $b$ then $b=0$ My Answer Assume that $$\lim_{x \to \infty}f(x)=L$$ then $$\lim_{x \to ...
1
vote
3answers
30 views

Where does my proof of uniform continuity fail?

I am trying to prove that $f:R \to R f(x)=\sin x$ is uniformly continuous. I have said: Fix $\epsilon > 0$ and $\delta=\epsilon$ $|\sin x - \sin y| \le |\sin x| - |\sin y| \le 1 - 1 = 0 ...
1
vote
1answer
58 views

Prove that any unbounded sequence has a subsequence that diverges to $∞$.

To prove that any unbounded sequence has a subsequence that diverges to ∞, is it enough to say that you can take a subsequence $(a_{m(k)})$ where $m(k)=k$, as you know that this diverges to infinity, ...
0
votes
2answers
38 views

I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
1
vote
1answer
14 views

Limit of a function proof verification

My proof: By Bernoulli Equation $(a^n+b^n)^{1/n}=b(1+(na)/b)^{1/n}$ By definition of a limit, fix $\epsilon > 0$ and $N>(b\epsilon^n)/a$ Then, $|a_n - b | = ...
2
votes
0answers
28 views

Proof verification-density of smooth compactly supported functions

I am trying to show that $C_{c}^{\infty}(\mathbb{R})$ (smooth compactly supported functions) is dense in $C_{c}(\mathbb{R})$ (in the $L^{p}$ sense). Can anyone check if my proof is correct? Let $f ...
4
votes
1answer
32 views

Prove there exists a point $c$ such thst $f(c)=c$ for the following function

If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function with $f(0)=2$ and $|f'(x)| \leq 1/2$ for all $x$ then there is a point $c$ such that $f(c)=c$ . My Attempt Let ...
1
vote
2answers
24 views

Series, limits and convergence.

Theorem $\,\bf3.3.1.\;$ If the series $$\sum_{n=1}^\infty a_n$$ is convergent then $\lim\limits_{n\to\infty}a_n=0$. Proof. Let $s_n=\sum_{k=1}^n a_k.$ Then by the definition the limit $\lim_n ...
1
vote
0answers
22 views

Show that both these Prove $f$ is differentiable from the right at $0$

Let $f:[0,1) \rightarrow \mathbb{R}$ be continuous on [0,1) and differentiable on $(0,1)$. Suppose the limit $\lim_{x \to +0} f'(x)$ exists. Prove that $f$ is differentiable from the right at $0$. ...
0
votes
2answers
30 views

Power Series and their radii of convergence

Suppose that $$\sum\limits_{n=0}^\infty a_nx^n \ \ and \ \ \sum\limits_{n=0}^\infty b_nx^n$$ $R$ and $S$ respectively.Let $U$ be the radius of convergence of $$\sum\limits_{n=0}^\infty c_nx^n$$ ...
1
vote
1answer
29 views

Convergence and uniform convergence of a sequence of functions

Suppose that ($f_n$;$n \geq 1$) is a sequence of functions defined on an interval [a,b].We will say $f_n$ tends to $f$ uniformly on $[a,b]$ as n tends to infinity if for every $\epsilon>0$ there ...
3
votes
0answers
35 views

Prove the following expression is true.

Let $x_1,...,x_{n+1}$ be arbitrary points in $[a,b]$ and let $$Q(x)= \prod\limits_{i=1}^{n+1} (x-x_i)$$Now suppose $f$ is an n times differentiable function and tha P is a polynomial function of ...
1
vote
1answer
13 views

Proof about First order derivative

Show that if $f'(c)>0$ then there exists $\delta>0$ such that $x \in (c,c+\delta) \ \ \implies \ \ f(x)>f(c)$ $x \in (c-\delta,c) \ \ \implies \ \ f(x)<f(c)$ My Attempt Now ...
2
votes
6answers
192 views

Showing that $\cos(z)$ has an essential singularity at $\infty$

Problem: Show that $\cos(z)$ has an essential singularity at $\infty$. EDIT: I just realized that step (2) is definitely wrong, as both those limits are undefined. Still, the sum of two ...
0
votes
2answers
86 views

If $f,g$ are uniformly continuous prove $f+g,fg$ are uniformly continuous

Suppose $f:E \rightarrow \mathbb{R}$ and $g:E \rightarrow \mathbb{R}$ are uniformly continuous, where $E$ is a subset of $\mathbb{R}$. Show that $f+g \ \ and \ \ fg$ are uniformly contiuous, what ...
1
vote
1answer
29 views

Show if $f$ is increasing then so is $f^{-1}$

Show if $f: \mathbb{R} \rightarrow \mathbb{R}$ is increasing then so is $f^{-1}$. My Attempt If $f$ is increasing then by definition; $$x<y \iff f(x)<f(y)$$ which implies ...
1
vote
1answer
31 views

One sided limits equal to actual limit

Suppose $f:(a,b) \backslash \{c\} \rightarrow \mathbb{R}$ is a function such that $$\lim_{x \to \ c+}f(x) \ \ \ \ and \ \ \ \ \lim_{x \to \ c-}f(x)$$both exists and are equal to a common value ...
0
votes
1answer
21 views

Using Properties of a Dense Set to prove characteristics of a continuous function

1. If $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous and $f(x)=0$ for all $x$ in a dense set $E$, then $f(x)=0$ for all $x \in \mathbb{R}$ 2. If $f:\mathbb{R} \rightarrow \mathbb{R}$ and ...
0
votes
1answer
17 views

For what points $c$ in $\mathbb{R}$ is $f$ continuous?

Let $X \subset \mathbb{R} $ be a fintie set and define $f:\mathbb{R} \rightarrow \mathbb{R}$ by $f(x)=1$ is $x \in X$ and $f(x)=0$ otherwise. At which points $c \in \mathbb{R}$ is $f$ continuous? ...
2
votes
0answers
81 views

Proof of uniform continuity on compact sets

Show that a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous on a compact set $K$ is uniformly continuous on $K$. Is the proof below correct? Proof: Let $\epsilon > 0$ and let ...
0
votes
1answer
28 views

If F is real-entire, then how to write, $F(z)- F(w)$ in terms of $(z-w)$ and $(\bar{z}- \bar{w})$?

Define $F:\mathbb C \to \mathbb C$ such that $F(z)= \sum_{j,k=0}^{\infty}c_{j,k} z^{j} \bar{z}^{k}$ is an entire real analytic function on $\mathbb C$ with $F(0)=0.$ My question is :How to show: ...
4
votes
1answer
30 views

Compact operator on invariant subspace is compact

Statement: Let $T \in \mathscr{B}(\mathscr{H})$, where $T$ is a compact operator. Let $M$ be a closed invariant subspace of $T$. Show that the restriction of $T$ to $M$ is compact. Attempted Proof: ...
0
votes
0answers
48 views

Using Cauchy's integral formula to find the best estimate for $|f^{(n)}(0)|$ under a condition

Question: Is the following proof valid? Ahlfors: If $f(z)$ is analytic for $|z| < 1$ and $|f(z)| \le 1/(1-|z|)$, find the best estimate of $|f^{(n)}(0)|$ that Cauchy's inequality will yield. ...
6
votes
0answers
59 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...
0
votes
1answer
51 views

Cauchy's theorem in a disk (Proof Verification)

Consider the following proof of Cauchy's theorem in a disk. My question is pasted at the bottom of the picture. (Note that in the proof below, a reference is made to "Theorem 2". In my textbook ...