6
votes
1answer
63 views

Alternative proof for the fact that a continuous function on a closed interval attains its boundaries.

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. We are interested in showing that $\exists \beta \in [a,b]$, such that $f(\beta) = M$, where M is its upper boundary. I have managed to proof ...
2
votes
1answer
37 views

Two question about how to compute this integral limit

Let $f: (-\pi,\pi]\to \mathbb R$ be continuous and let $p_x (u) = {(f(u+x) - f(x)) \cos ({u \over 2}) \over \sin ({u \over 2}) }$. I want to show that $$ \int_{-\pi}^\pi p_x(u) \sin (Nu) du \to 0$$ ...
3
votes
0answers
48 views

If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$ f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N. $$ Suppose there exists a continuous function $g$ on ...
2
votes
1answer
76 views

Checking proof that $f(x)=x^2+1$ is continuous

Let $f:\mathbb R \to \mathbb R$ is defined by $f(x)=x^2+1$. Prove this function is continuous for all $x \in\mathbb R$. Here is what I have: Suppose that $c∈ℝ$. Let $\varepsilon>0$. Let ...
1
vote
1answer
34 views

Operator $Au(t) = \int_0^t e^{t-s} u(s) ds$ (Proof Verification)

Consider the space $C([0,1])$ with $||\cdot||_\infty$ norm. Let $A: C([0,1])\rightarrow C([0,1])$ be the operator defined by $$Au(t) = \int_0^t e^{t-s} u(s) ds.$$ And I am not 100% sure about (c), ...
1
vote
3answers
31 views

Considering $\epsilon$ intuitively in limit proof

I'm having rather difficult time in trying to use $\epsilon$ argument appropriately. For example here is my simple $\epsilon$ proof in one question. The question is as follow: Prove if $s_n \geq 0$ ...
1
vote
1answer
52 views

Is my proof on showing that the set of monotone functions on $[a,b]$ has cardinality of continum correct?

I was given an exercise problem to show that the cardinality of the set of all monotone functions on $[a,b]$ is $\aleph$. I came out with a proof which I am not sure if it is correct. My proof: Let ...
3
votes
0answers
33 views

“Simple” question (Lebesgue Integration) in a hard exam (proof verification)

Let $f_n\in L^1(0,1)$ and $C>0$ be such that $f_n \geq 0, f_n \rightarrow 0$ a.e., and $$\int_0^1 \max\{f_1, ..., f_n\} dx \leq C \quad \text{ for every } n.$$ Prove that $f_n \rightarrow 0$ in ...
1
vote
1answer
23 views

How can I show that a r.v. with cumulative distribution is continuous?

I want to show that, if $F_X$ is the cumulative distribution function of a random variable $X$, then $X$ is absolutely continuous iff $F_X \in C^1(\mathbb{R})$ ? I know absolutely continuous means ...
1
vote
1answer
40 views

$B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\cap_k B_k$ is either a point or a closed ball.

$B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\bigcap_k B_k$ is either a point or a closed ball. Please help me check the proof, thanks! Define $x_k$ to be ...
1
vote
1answer
106 views

Show there exists $x\in (0,1)$ such that $f(x) \leq \int_0^1 f(t) dt$

Please help me check my proof, thanks! (a) Show there exists $x\in (0,1)$ such that $$f(x) \leq \int_0^1 f(t) dt.$$ Proof: when $f$ is constant a.e, the equality holds for all points except for a ...
2
votes
1answer
49 views

Proof: $(\sup(A) - \epsilon)^n<y<(\sup(A)+\epsilon)^n$

Prop.: let be $y \in \Bbb{R}_{>0}$, $n \in \Bbb{N}_{>0}$, and $A \subseteq \Bbb{R}$, then: $$A=\{x| x \in \Bbb{R}_{>0}\wedge x^n \leq y \} \Rightarrow (\sup(A) - \epsilon)^n< ...
6
votes
1answer
49 views

$\int_0^1 (f(x))^n =$ constant, $f\geq 0$, then $f$ is a characteristic function of a measurable set.

$\int_0^1 (f(x))^n =$ constant, $f\geq 0$, then $f$ is a characteristic function of a measurable set. This is the result from question part (a). Now for part (b), will it also hold when the ...
5
votes
2answers
138 views

Is my proof that $\lim\limits_{n\to +\infty}\dfrac{u_{n+1}}{u_n}=1$ correct?

I'm doing an exercise where $(u_n)$ is a numerical sequence which is decreasing and strictily positive.While $(u_n)$ is a numerical sequence which is decreasing and strictily positive, then $(u_n)$ is ...
1
vote
1answer
72 views

Proof about Riemann integrability of a bounded function

I tried to prove the following, please could somebody tell me if my proof is correct? If $f: [a,b]\to \mathbb R$ is a bounded Riemann integrable function then for every $\varepsilon > 0$ there ...
0
votes
1answer
80 views

Check proof please

Prove that if: $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ than: $\lim_{x\rightarrow\infty}{\frac{f(x)}{x}}=L$ Assuming $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ we can choose $X_{\epsilon}$ s.t. ...
0
votes
1answer
62 views

Term by term integration

Let $D \subset \mathbb{R}^{d} $ be open. For $u,v \in C_{0}^{\infty}(D)$, we define \begin{eqnarray*} \mathcal{A}(u,v)=\sum_{i,j=1}^{d} \int_{D} \frac{\partial u(x) }{\partial x_{i}}\frac{\partial ...
0
votes
0answers
16 views

Proof verification related to the discrete metric

Can someone please verify my proof? Let $X_1$ be a set and let $d_1$ be the discrete metric on $X_1$. (a) Prove that every subset of $(X_1, d_1)$ is open. (b) Prove that if $(X_2, d_2)$ ...
2
votes
0answers
22 views

Prove that $d_\infty(f, g) = \operatorname{sup}\{|f(x)-g(x)|:x \in [a,b]\}$ defines a metric

Can someone please verify my proof? Let $C[a,b]$ denote the set of all continuous functions from $[a,b]$ to $\mathbb{R}$. Let $d_\infty:C[a,b] \times C[a,b] \longrightarrow [0, \infty)$ be given ...
2
votes
2answers
73 views

Proving Lebesgue integration result

I have a Lebesgue integration question and a proposed proof. Please advise. Let $\Omega \subset \mathbb{R}^{n}$(denote the boundary as $\partial \Omega$) and consider $$\int_{\partial \Omega} vf ...
1
vote
1answer
61 views

Characteristic function of Cantor set is Riemann integrable

I want to prove that the characteristic function of the Cantor set is Riemann integrable on $[0,1]$. Could somebody please tell me if my proof is correct? Let $f$ be the characteristic function of ...
3
votes
1answer
40 views

Convergence of measures — revisited

In this thread, I asked a question about the convergence of measures. The conjecture I posed there, which turned out to be false, was supposed to be a lemma that I wanted to use to prove a ...
0
votes
1answer
27 views

Unique solution for $\int_x^1 f(t) dt = 2x$ and $|x| < \epsilon$

Let $f$ be continuous on $\mathbb{R}$ such that $$f(0) \neq -2 \quad\text{ and } \quad \int_0^1 f(t) = 0.$$ Show that there exists $\epsilon > 0$ such that the equation $$\int_x^1 f(t) dt = 2x$$ ...
0
votes
1answer
39 views

Show $f(t) = t$ given $\int_0^1 t^n f(t) dt = \frac{1}{n+2}\quad \forall n\in\mathbb{N}.$

Let $f$ be in $L^2(0,1)$, show that $f(t) = t$ if and only if $$\int_0^1 t^n f(t) dt = \frac{1}{n+2}\quad \forall n\in\mathbb{N}.$$ To show $\Leftarrow$ Let $g(t) = f(t) - t$, then $$\int_0^1 t^n ...
2
votes
0answers
52 views

$f\in L^1(0,\infty)$ monotone, show $\lim_{x\rightarrow \infty} xf(x) = 0$ [duplicate]

Here is the solution: First $f$ is monotone and integrable on $(0,\infty)$, wolg we can assume that $f>0$ and approaches $0$ as $x$ goes to infinity. Observe that $$xf(2x) \leq \int_x^{2x} ...
0
votes
2answers
113 views

Proof Verification: $ (ab)^{-1}=a^{-1}b^{-1}$

I am asked to prove $$(ab)^{-1} = a^{-1}b^{-1}$$ where $a,b\in\mathbb{R}\setminus\left\{0\right\}$. Here is what I have: $$(ab)^{-1} = \frac{1}{ab} = \frac{1}{a} * \frac{1}{b} =a^{-1}b^{-1}$$ ...
0
votes
1answer
25 views

Proving the Nested Interval Property using Axiom of Completeness

I'm self-studying real analysis using Abbott's text "Understanding Analysis." I'm trying to think out/prove as much on my own as I can, so I am working on proving the Nested Interval Property (Theorem ...
3
votes
1answer
75 views

Changing one point does not change the Riemann integral

I tried to prove the following. Please could somebody tell me if my proof is correct? Let $f: [a,b]\to \mathbb R$ be Riemann integrable. Then changing one value of $f$ then $f$ is still ...
0
votes
2answers
34 views

Vanishing moments and integrability

Is this correct? $\int_\mathbb{R}x^m f(x) dx=0 \iff \int_\mathbb{R}x^m \overline{f(x)}\,dx =0$. If yes then please tell the conditions under which this holds.
2
votes
1answer
64 views

Alternative definition of complex number, showing it is equivalent to the tradidional one.

The author of a book makes an alternative definition of the complex numbers, later he shows that this definition is equivalent to the ordinary definition where we define $i^2=-1$. Here is his ...
2
votes
2answers
82 views

Proof that the limit of $\frac{1}{x}$ as $x$ approaches $0$ does not exist

Hello I was hoping that someone might be able to verify that the following proof that $\lim_{x\to 0} {1\over x}$ does not exist is correct. First assume that $\lim_{x\to 0} {1\over x} = L$. This ...
0
votes
0answers
22 views

Limit and uniform convergence related proof

Can someone please verify this Let $f$ be a real valued function on (0, 1). Define a sequence of functions as $$f_n(x) = \left\{ \begin{array}{ll} \alpha & x < \frac{1}{n} \\ ...
2
votes
3answers
72 views

$x^{1+\epsilon}$ is not uniformly continuous on $[0,\infty)$

There are two questions. First: is the proof underneath correct? Let $\epsilon>0$ and let $f(x)=x^{1+\epsilon}$. I aim to show that $f$ is not uniformly continuous on $[0,\infty)$. We will show ...
3
votes
1answer
35 views

Showing that for $s,t\in\mathbb{Q}$, we have $(s+t)^*= s^* + t^*$.

I'm working through the problems of Elementary Analysis Theory of Calculus, and for some reason, this question didn't make the solutions in the back of the book. I did a thorough search on Stack ...
3
votes
1answer
44 views

Can a proper Morse function $\mathbb{R}\to\mathbb{R}$ have infinitely many critical points?

Depending on interpretation, there may be an assumption missing from Exercise 6.1.4(a) in Liviu I. Nicolaescu's Invitation to Morse Theory: Suppose $f : \mathbb{R} → \mathbb{R}$ is a proper Morse ...
0
votes
1answer
24 views

Convergence of the maximum of a sequence of functions which converge uniformly on a closed interval

Can someone please verify this? Let $f_n$ be a sequence of continuous functions on a closed interval $I$ converging uniformly to $f$. Is it true that max $\{f_n(x):x\in I\}$ converges to max ...
1
vote
2answers
29 views

Proof Check: Every Cauchy Sequence is Bounded

Sorry if I keep asking for proof checks. I'll try to keep it to a minimum after this. I know this has a well-known proof. I understand that proof as well but I thought I'd do a proof that made sense ...
2
votes
1answer
51 views

If the integral of a non-negative function is $0$, then the function is $0$

Suppose that $f$ is a continuous function on $[a,b]$ and that $f(x)\geq0$ for all $x\in [a,b]$. Show that if $\int_a^bf(x)=0$, then $f(x)=0$ for all $x\in[a,b]$. Let $F(x)=\int_a^xf(x)$. Since ...
2
votes
1answer
33 views

Convergence of similar power series given a convergent series

Can someone verify this? Suppose that the series $$\sum\limits_{n=1}^\infty a_n x^n$$ has a radius of convergence $R$, where $0 < R < \infty$ (a) Find the radius of convergence of ...
5
votes
1answer
62 views

Uniform convergence of $\sum\limits_{n=1}^\infty \sin \left(\frac{x}{n^2}\right)$

Can someone please verify my answers? Consider the series $$\sum\limits_{n=1}^\infty \sin \left(\frac{x}{n^2}\right)$$ Prove that the series converges uniformly on the bounded interval $[-M, ...
0
votes
1answer
42 views

Show that $(\log |x|)^2\notin \text{BMO}([-1,1])$.

I am trying to show that $u(x)\equiv (\log |x|)^2\notin \text{BMO}([-1, 1])$ by showing that it doesn't satisfy the John-Nirenberg inequality. If $u\in\text{BMO}[-1, 1])$ then this inequality says ...
0
votes
2answers
52 views

Existence of $\delta$

Exercise Assume that $K$ and $A$ are disjoint nonempty subsets of $ \Bbb{R}^n$ with $K$ compact and $A$ closed. Prove by using wat we know about $d(x,A)$ that there exists $\delta >0$ such that ...
2
votes
1answer
34 views

$\varepsilon$-$\delta$ proof of continuity of floor function $\lfloor x\rfloor$

I would just like to ask someone to confirm or correct the following 'proof' of continuity of the floor function. Let $\varepsilon>0$ be given. Set $\delta:=\min\lbrace x-\lfloor x\rfloor,\lceil ...
0
votes
3answers
28 views

Differentiability of $f(x) = x \sin \frac{1}{x}$ for $x \neq 0$ and $0$ for $x = 0$.

Can someone please verify this (I admit the proof is very terse, but is the reasoning correct)? Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be defined by $f(x) = x \sin \frac{1}{x}$ for $x \neq ...
4
votes
1answer
71 views

Proof verification: $\int_a^x f(t) \text{dt}=0$, $f$ is continuous at $x$. Prove that $f(x)=0$

Let $f:[a,b]\to R$ be an integrable function such that for all $x \in[a,b]$, we have $\int_a^x f(t) \text{dt}=0$. Show that if $f$ is continuous at $x \in [a,b]$, then $f(x)=0$. My attempt: argue ...
1
vote
1answer
31 views

Uniform convergence of $f_n(x) = \frac{nx}{1+nx^2}$

Can someone please verify my proof? Let $f_n(x) = \displaystyle{\frac{nx}{1+nx^2}}$ (a) Find $f(x) = \lim f_n(x)$ (b) Does $f_n \longrightarrow f$ uniformly on $[0,1]$? (c) Does ...
2
votes
1answer
28 views

Uniform convergence of $f_n(x)=\left(x-\frac{1}{n}\right)^2$ for $x \in [0,1]$

Could someone please verify this? Let $f_n(x)=\left(x-\frac{1}{n}\right)^2$ for $x \in [0,1]$. (a) Does the sequence $(f_n)$ converge pointwise on the set $[0,1]$? If so, give the limit ...
2
votes
5answers
53 views

Please check my proof on “$f(x)=\frac{1}{x}$ is not uniformly continuous on $(0,\infty)$”

$f(x)=\frac{1}{x}$. Prove that $f$ is not uniformly continuous on $(0,\infty)$. We want to find $\epsilon_0 >0$ such that $\forall \delta>0$, there are some $x,y \in (0,\infty)$ such that ...
0
votes
1answer
33 views

Would like to compute the limit of some integral

I was working on a exercise where the goal was to compute the following limit, $$\lim_{n\rightarrow\infty}\int_{\mathbb R}e^{-|x|n}e^{-\frac{x^2}{2}}dx$$ and some tutor of mine claimed that the limit ...
2
votes
2answers
53 views

Suppose $a_n \geq 0$, and $\sum a_n$ diverges, and $\lim a_n = 0$. Show that $\sum \frac{a_n}{1+a_n}$ diverges.

Can someone please verify my proof sketch? Suppose $a_n \geq 0$, and $\sum a_n$ diverges, and $\lim a_n = 0$. Show that $\displaystyle{\sum \frac{a_n}{1+a_n}}$ diverges. Let $\epsilon > 0$. ...