0
votes
1answer
98 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
453 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
40 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
66 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
72 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
43 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
32 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
29 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
46 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
37 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
26 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
30 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
23 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
21 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
12 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
188 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
80 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
29 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
18 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
76 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
26 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
47 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
52 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
49 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 ...
1
vote
1answer
49 views

Understanding why $\int_\gamma {dz \over z - a} = k 2\pi i$ for $\gamma$ a closed curve not passing through $a$

The following is a paraphrased proof from Ahlfors. I bolded the part that is confusing me and asked a question about it at the bottom of this post. Hypothesis: Let $\gamma$ be a closed curve that ...
3
votes
2answers
64 views

Analytic $F(z)$ has $f(z)$ as derivative $\implies$ $\int_\gamma f(z)\ dz = 0$ for $\gamma$ a closed curve

Hypothesis: Suppose that $F(z)$ has $f(z)$ as a derivative. Suppose further that $F(z)$ is analytic. Now consider the complex line integral $$ \tag{1} \int_\gamma f(z)\ dz $$ Question: Does this ...
0
votes
1answer
29 views

Verifying a condition for which $\int_\gamma p\ dx + q\ dy$ depends only on endpoints

Hypothesis: Suppose there exists a function $U(x,y)$ in $\Omega$ with partial derivatives $${\partial U \over \partial x} = p \quad \quad {\partial U \over \partial y} = q$$ Goal: Show that the ...
1
vote
0answers
23 views

Computing a complex line integral $dz$ in terms of line integrals $dx$ and $dy$

Goal: I'm trying to verify the calculation claimed by Ahlfors that $$\int_\gamma f(z)\ dz = \int_\gamma (u\ dx - v\ dy) + i \int_\gamma (u\ dy + v\ dx)$$ Attempt: $$\int_\gamma (u\ dx - v\ dy) + i ...
3
votes
1answer
49 views

Some questions about proof of Theorem 2.43 in Baby Rudin

I will include the proof here and highlight the parts that are giving me trouble. Theorem $\hspace{5 pt}$ Let $P$ be a nonempty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable. Proof ...
1
vote
0answers
24 views

A simple proof with directional derivatives

Suppose $\nabla f (x) \cdot d < 0$, prove that there exists $\delta > 0$ such that $$f(x + \tau d ) < f(x)$$ for all $\tau \in (0, \delta)$ My proof consists of only a few lines. ...
0
votes
2answers
37 views

If f is continuos on an interval, is it then uniformly continuous

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I now know that it is not. Can someone give me a proof ...
0
votes
1answer
26 views

If $f$ is continuous on $(0,5)$, is it uniformly continuous on same interval

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I believe it is. I now know that it is not. Can someone ...
0
votes
1answer
39 views

Holder's Inequality Proof Verification

Wikipedia outlines a nice proof of Holder's Inequality in the link provided. The fifth sentence in the proof reads: Dividing $f$  and $g$ by $\|f \|_p$ and $\|g\|_q$, respectively, we can assume ...
2
votes
0answers
35 views

Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) dx = 1$ for $t>0$.

Suppose $f \in \mathcal{R}$ on $[0,A]$ for all $A < \infty$, and $f(x) \rightarrow 1$ as $x \rightarrow + \infty$. Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) ...
2
votes
1answer
167 views

Proof concerning equivalent definition of supremum using limits

I believe I have a proof to the following theorem: Let $S$ be a subset of $\mathbb{R}$ that is non-empty and bounded above. $s \in \mathbb{R}$ is the supremum iff $s$ is an upper bound of $S$ and for ...
1
vote
1answer
27 views

Verify why this is not a metric $d(x,y)=\|x-y\|_p$ ( $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$).

$d(x,y)=\|x-y\|_p$ $p$ is prime and $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$, where $m, n$ are coprimes with $p$. This is not a metric because if $x=y=p^k\dfrac{m}{n}$, then $x-y=0=p^0\dfrac0n$. ...
1
vote
1answer
75 views

$v$ is Conjugate Harmonic to $u$ $\implies$ $f = u + iv$ is Analytic (Proof Verification from Ahlfors)

Hypothesis: Let $u$ and $v$ be two functions from $\mathbb{R}^2$ to $\mathbb{R}$ s.t. $$ \Delta u = {\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} = 0 $$ and $$ \Delta v = ...
2
votes
4answers
76 views

A question about metrizability

In a lecture in Topology I had earlier this week, I was told (without proof) that not every topological space $(X,O)$ is metrizable, i.e, it is impossible to find some metric $d$ such that $O$ and ...
4
votes
2answers
39 views

Orthogonal representation of finite operator

I would like to know if my proof is correct. Statement: Let $T$ be a finite rank operator on a Hilbert space $\mathscr{H}$. Show that $\forall \, h \, \in \mathscr{H}, \, T(h)$ can be written as ...
5
votes
0answers
93 views

$f$ continuous nowhere implies $f$ not Riemann integrable

I am trying to prove the statement in the title, without using the theorem which says that the set of discontinuities of a Riemann integrable function must have measure $0$. I am aware that similar ...
0
votes
1answer
57 views

Isometry is not surjective

According to the definition I am using, an isometry is a mapping $f:X \rightarrow Y$ between two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$: $$ d_{Y}(f(a),f(b)) = d_{X}(a,b) $$ for all $a,b \in X $ I ...
1
vote
1answer
67 views

Possible book correction or am I missing something?

Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y ...
0
votes
1answer
52 views

Showing that $\log \log(z)$ is Analytic (Proof Verification)

Goal: Convert $\log \log (z)$ into a single-valued function defined on a suitable region of $\mathbb{C}$ and then prove that it is analytic. Attempt: As has been demonstrated elsewhere, we have ...