Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

2
votes
1answer
21 views

If $F \subseteq \mathbb{R}$ is closed, countable infinity. Prove that $F$ has a countless infinite number of isolated points.

To prove it use the following lemma : If $F$ is closed and $x \in F$ then $x$ is a isolated point of $F$ if and only if $F-\{x\}$ is closed. In a book, a solution is as follows : Let's suppose ...
0
votes
1answer
58 views

Integral of $fg$ over area $A$ vanishes for every $g \Rightarrow f = 0$

I am asked to produce a rigorous proof of the following: Let $A$ be some area in $\mathbb{R}^2$ whose boundary $\partial A$ is smooth, let $g$ be a $C^2$ function on $\bar A$ (the closure of $A$) ...
1
vote
2answers
53 views

Proving that a particular set in $\mathbb R^2$ is open.

I have to prove that the subset $A=\{ (x,y)|x^2>y\}$ of $\mathbb R^2$ is open. I graphed it, and I can see that it is open, and I can also see that the boundary is $y=x^2$. We have not yet ...
0
votes
1answer
33 views

How do I prove that $(2n+1)!! \gt t^{2n+2}$ as $n \to \infty$ for all $t \in \mathbb R$?

I would like to prove the following conjecture: $$\lim_{n \to \infty} \frac{t^{2n+2}e^{-\frac{1}{2}t^2}}{(2n+1)!!}=0$$ for all $t \in \mathbb R$. Naturally, this can only be done by showing that $(2n+...
0
votes
1answer
21 views

Is $h(x)=x^\frac{3}{2}$, $D_h=[0,\infty)$ uniformly continuous?

Is $h(x)=x^\frac{3}{2}$, $D_h=[0,\infty)$ uniformly continuous ? My attempt: i) since $h$ is continuous on $[0,2]$ then it is uniformly continuous on $[0,2]$. ii) If $x,u \in [1,\infty)$, then $|...
1
vote
0answers
25 views

Problem about weak convergence using continuous bounded functions

The question is from Billingsley. $X_n \in \{\gamma_n+k\delta_n; k\in N\}, \delta_n>0$. Suppose $\delta_n\rightarrow 0$ and $k_n$ is an integer varying with n s.t. $\gamma_n+k_n\delta_n\rightarrow ...
0
votes
2answers
19 views

Sup norm. Is the integral of functions always greater than or equal to the integrand? [duplicate]

I am doing some proofs in real analysis with the sup-norm metric. In my arguments, I have that $sup\: x\in[0,1] \big\{\int_{0}^{x}|f(t)-g(t)|dt\big\}\leq sup\: x\in[0,1] \big\{|f(t)-g(t)|\big\}$. My ...
0
votes
1answer
26 views

Non-zero function $f(x)$ such that $\int_{1}^{\infty}\frac{f(\frac{1}{t})}{t}dt-\sum_{k=1}^{\infty}k\int_{k}^{k+1}\frac{f(\frac{1}{t})}{t^2}dt=0$

Does there exists a function $f(x)\in L^2(0,1)$, not identically zero such that $$\displaystyle \int_{1}^{\infty}\frac{f(\frac{1}{t})}{t}dt-\displaystyle \sum_{k=1}^{\infty}k\int_{k}^{k+1}\frac{f(\...
-1
votes
1answer
23 views

Draw and study the discontinuous of $g(x)=\lfloor \sin(x) \rfloor$

Draw and study the discontinuous of $g(x)=\lfloor \sin(x) \rfloor$ $$g(x)=\begin{cases} -1 & x\in (-\pi,0) \\ 0 & x \in [0,\pi]\setminus \lbrace \frac{\pi}{2} \rbrace \\ 1 & x=\frac{\pi}{...
7
votes
4answers
105 views

If $‎\displaystyle{\lim_{x\to\infty}}‎\frac{f(x)}{g(x)} = 1‎$,‎ then $‎\displaystyle{\lim_{x\to\infty}}(f(x) - g(x)) = 0‎$.

‎Suppose ‎$‎f‎$ ‎and ‎‎$‎g‎$‎ ‎‎are real functions such that $‎‎‎\displaystyle{\lim_{x\to\infty}}‎\frac{f(x)}{g(x)} = 1‎$‎‎. ‎My ‎question ‎is‎: ‎‎‎ ‎‎What other condition is required ‏‎that $‎‎‎\...
0
votes
0answers
41 views

Problem in my final solution of a proof regarding inequalities

We are given the following problem. Assume that $\epsilon < \int_0^1 f(x)dx < \epsilon$ and $-\epsilon < f'(x) < \epsilon $ for all $x \in [0,1]$ Now show that this leads to $f(0) - \...
0
votes
0answers
35 views

Discuss the convergence of a sequence [duplicate]

If $k, x_1$ are positive, and $x_{n+1}$ = $\sqrt {k + x_n},$ discuss the convergence of the sequence $x_n$, according to whether $x_1$ is less than or greater than $\alpha$, the positive root of $x^2 =...
1
vote
1answer
46 views

Without use of derivatives, prove that function $(1+x^p)^{1/p}$ is convex for $p\geq 1$

Without use of derivatives, prove that function $(1+x^p)^{1/p}$ is convex for $p\geq 1.$ Attempt. The result is obvious for $p=1$, since function $x+1$ is affine. For $p>1$, functions $1,~x^p$ are ...
0
votes
0answers
30 views

Verify that $f(x)=\frac{1}{x}-\frac{1}{x_0}$ is continuous for every $x_0\neq 0$.

Verify that $f(x)=\frac{1}{x}-\frac{1}{x_0}$ is continuous for every $x_0\neq 0$. $f(0)$ is not defined. So the function is discontinuous at $0$. Let $c\in \mathbb{R}\setminus \lbrace 0 \rbrace$, we ...
1
vote
2answers
78 views

Find $\lim_{n\rightarrow\infty}\sqrt[n]{5+n^2}$ using $\varepsilon-N$ language

Use an $\varepsilon-N$ argument to find and prove $\lim_{n\rightarrow\infty}\sqrt[n]{5+n^2}$. Try some variations of your own. I think that the limit is $1$, since the limit as $n$ tends to infinity ...
0
votes
0answers
19 views

Find extrema and roots of a function from its Fourier series expansion

That might be a dumb question, but here it comes. Say we have a function defined by its fourier series expansion as follows : $$\forall x\in\mathbb{R}, f(x)=\sum\limits_{n=1}^{\infty}A_n\cos\left(2\...
0
votes
2answers
40 views

A sequence with infinite number of limit points.

In my real analysis class, we have to determine whether or not the following is true. "There exists a sequence of real numbers that has infinite number of limit points." It then seemed to be true and ...
0
votes
0answers
31 views

Help proving this map is closed

Define the following equivalence relation on $\mathbb{R}^2$: $(x,y)\sim(x',y')$ iff there is $n\in \mathbb{Z}:(x',y')=(x+n,(-1)^ny)$. Let be $E=\mathbb{R}^2/\sim$ the quotient space, and $q:\mathbb{R}^...
0
votes
0answers
26 views

Upper and lower bounds for a finite sum [duplicate]

Find upper and lower bound for the following finite sum: $\frac{1}{1} + \frac{1}{2^3} + \frac{1}{3^3} + ··· + \frac{1}{n^3}$ My attempt: Using the integral test: we know that $\frac{1}{1} + \frac{...
1
vote
2answers
58 views

Direct proof of Archimedean Property (not by contradiction)

I looked at the proof of Archimedean Property in several places and, in all of them, it is proven using the following structure (proof by contradiction), without much variation: If $\space x \in \...
1
vote
0answers
22 views

Linear convergence of sequence

I have the following exercise but it seems to me that this is false : Let $(x_n)_{n \in \mathbb{N}}$ by a sequence of real numbers and $x^* \in \mathbb{R}$. We say that the sequence $(x_n)$ ...
-1
votes
1answer
39 views

uniform convergence and pointwise of $\sum_{n=0}^{+\infty} \frac{n+1+(-1)^n}{(n+1)!} x^n$ [on hold]

The radius of convergence is R=1 so there is pointwise convergence in $(-1,1)$. Had I to study also $x=1,-1$? If I compute the sum had I to descern pair and odd cases: $\sum_{n\ge 0}\frac{n}{(n+1)!}+...
2
votes
2answers
57 views

It says to prove that the closed ball in $\mathbb{R}^2$ with the Euclidean metric is this, but isn't that the definition of a closed ball?

Prove that in $\mathbb{R}^2$ $$\overline{B_1((0,0))}=\{x\in \mathbb{R}^2:d_2(x,0) \leq 1\}$$ where $$d_2 (x,y) = \sqrt{\sum_{i=1}^n(x_i-y_i)^2}$$
0
votes
0answers
32 views

Upper and Lower bound of a finite sum

Find upper and lower bound for the following finite sum: $$\frac{1}{1} + \frac{1}{2^3} + \frac{1}{3^3} + ··· + \frac{1}{n^3} $$ My attempt is: Using the integral test: we know that $\frac{1}{1} + ...
0
votes
0answers
35 views

Show that the series $\sum_{n=1}^\infty{f_n}(x) = \frac{x}{n^\alpha(1+nx^2)}$ conv. uniformly over any interval $[a,b]$ where $\alpha>1/2$

My attempt included finding the maximum of the sequence of functions to find the maximum of the function and then using the Wierstrauss $M$-Test. However, I couldn't find a suitable $M$ that worked ...
0
votes
1answer
57 views

find the upper and lower bound for a finite sum

Find upper and lower bound for the following finite sum $1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ My attempt: $1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ = $\sum_{i=1}^n ...
0
votes
1answer
31 views

Is L infinity compactly embedded in L2?

When the measure space has finite measure is this true? I could take an equibounded sequence in L infinity, thus for almost every point i'd get a bounded sequence in the complex plane, build this way ...
1
vote
1answer
34 views

Help with convergence tests for series

I have a few questions to ask about series and convergence tests. I have been struggling to study everything fully and if someone can give me advices I will be really thankful.This is what I know so ...
3
votes
0answers
37 views

Convolution square of the Cantor set

For $0\leq d\leq 1$, let $\eta_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$ (for some normalization); recall that it is translation-invariant. Motivation for what follows: Up to ...
0
votes
2answers
50 views

If a function $f: A \to \mathbb R$ is bounded and continuous on $A$, does its integral on said interval converge uniformly?

Let the bounded interval $A$ be a subset of the reals and let $f : A\to\mathbb R$ be a bounded and continuous function. Since the sequence $(f_n): f_n = f/n$ has the limit $\displaystyle\lim_{n\to\...
0
votes
1answer
66 views

Prove $\int_{a}^{\xi}{f(t)}dt=\int_{\xi}^{b}{\frac{1}{f(t)}}dt$ [on hold]

Let $f(x)\in C[a,b],f(x)>0$,Prove that: $\exists! \xi\in(a,b)$ $$\int_{a}^{\xi}{f(t)}dt=\int_{\xi}^{b}{\frac{1}{f(t)}}dt$$ I don't have any idea about this,so I hope I can get some help or hint. ...
3
votes
1answer
35 views

$f(x) = \begin{cases} 0 & x\in( \mathbb{Q}\cap [0,1])^c \\ \frac{1}{q} & x=\frac{p}{q} \in \mathbb{Q} \cap [0,1], (p,q)=1 \end{cases}.$

let $f$ be the function defined by $$f(x) = \begin{cases} 0 & x\in( \mathbb{Q}\cap [0,1])^c \\ \frac{1}{q} & x=\frac{p}{q} \in \mathbb{Q} \cap [0,1], (p,q)=1 \end{cases}.$$ Prove ...
1
vote
1answer
27 views

Does the interval (0, 1] satisfy the least upper bound property?

I'm having difficulty understanding one of the very early theorems proved in Rudin, namely that the least upper bound property implies the greatest lower bound property. Why is the following not a ...
2
votes
2answers
47 views

If $f$ is convex and concave on interval $I$, then $f$ is affine on $I$

Prove that if $f$ is convex and concave on interval $I$, then $f$ is affine on $I$. Attempt. Although this topic has been discussed before, i would like to point out some technique issues. 1) Let $I=...
11
votes
2answers
253 views
+50

Proof (without use of differential calculus) that $e^{\sqrt{x}}$ is convex on $[1,+\infty)$.

Prove (without use of differentiation) that $f(x)=e^{\sqrt{x}}$ is convex on $[1,+\infty)$. Attempt. Function $x\mapsto e^x$ is convex and increasing, but $x\mapsto \sqrt{x}$ is concave, so we ...
0
votes
0answers
23 views

Compute $\sum_{n=1}^{+\infty} \frac{x^n}{n n!}$ [duplicate]

The radius of convergence of this series is $R=+\infty$ so the series converges in all R and uniform in all limited subset of R. To compute the sum I don't understand how do. I know $x^n=n\int_{0}^{...
3
votes
1answer
76 views

Solving an ODE system which depends on a periodic function

I have the following ODE system and want to show that all solutions $\gamma(t)=(x(t),y(t),z(z))$ exist for all times (or can be extended on all of $\mathbb{R}$). The only problem is that the system ...
2
votes
3answers
114 views

Prove $\lim_{n\to\infty}\frac{\int_{-1}^{1}{f(x)(1-x^2)^n}dx}{\int_{-1}^{1}{(1-x^2)^n}dx}=f(0)$

Prove that:Let $f\in C[-1,1]$ $$\lim_{n\to\infty}\frac{\int_{-1}^{1}{f(x)(1-x^2)^n}dx}{\int_{-1}^{1}{(1-x^2)^n}dx}=f(0)$$ My attempt: $$\begin{eqnarray}\lim_{n\to\infty}\frac{\int_{-1}^{1}{f(x)(1-x^...
1
vote
3answers
61 views

Prove that $\lim_{n\to\infty}\int_{0}^{1}\cos^n\left(\frac{1}{x}\right)\,dx=0$

Prove that:$$\lim_{n\to\infty}\int_{0}^{1}\cos^n\left(\frac{1}{x}\right)\,dx=0$$ I have a idea about this,but I can't complete proof. Hint: We can split the integral into two parts,then estimate ...
1
vote
1answer
37 views

Prove that for $f\in L^1(\mathbb R)$, $\int_{\mathbb R}|f|=0\implies f=0$ a.e.

Let $f\in L^1(\mathbb R)$ s.t. $$\int_{\mathbb R}|f|=0\implies f=0\ a.e.$$ My attempt Suppose $f$ continuous and that there is $y$ s.t. $|f(y)|\neq 0$. In particular, there is $\delta >0$ s.t. $f(...
6
votes
1answer
102 views

Interpretation of Differentials

$$ \newcommand{\qa}{P} \newcommand{\qb}{Q} \newcommand{\da}{dP} \newcommand{\db}{dQ} \newcommand{\positiverealnumbers}{\mathbb{R}_+} \newcommand{\realnumbers}{\mathbb{R}} \newcommand{\naturalnumbers}{\...
5
votes
1answer
48 views

Is a set of measure zero in $\mathbb{R}$ totally disconnected?

Let $M \subset \mathbb{R}$ be a nonempty set of Lebesgue measure zero. Does it follow that $M$ is totally disconnected in the sense that for any $x<y$, with $x,y\in M,$ there exists $z\notin M$ ...
6
votes
1answer
87 views

Doubt in Understanding Lebesgue measure.

I am studying Measure theory form Stein and Shakarachi:Real Analysis. I come across observation regarding the outer measure. For any $E\in R^d$ $m_*(E)=\inf m_*(O)$ where $O $ is the open set ...
2
votes
1answer
23 views

Contradiction in radii of convergence? Where is my error?

I'm working through Baby Rudin and I came across what seems to me to be a contradiction, but it could be an error on my part. It has to do with radii of convergence of power series. First, let $\{...
0
votes
0answers
39 views
+100

Shattering with sinusoids

Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$, does there exist an ...
1
vote
1answer
16 views

An Sobolev-type inequality of $1-d$ torus

In Kishimoto-Tsutsumi's paper published by Math Research Letter 2018, I see the following inequality in the last line of page 10: $\| f g \partial_x h \|_{H^{-s}(T)} \lesssim \| f \|_{H^s(T)} \| g \...
5
votes
1answer
39 views

Limit of $\lim \limits_{x \to\infty} (\frac{x}{1-x})^{2x}$

I've been struggling with this term for a while now: $$\lim \limits_{x \to\infty} (\frac{x}{1-x})^{2x}$$ I know it has to do something with $\lim \limits_{x \to\infty} (1+\frac{n}{x})^x=e^n$ but didn'...
2
votes
3answers
50 views

Let $f:X\to X $ be continuous. Does $f $ have a fixed point when $X=[0,1)$ or $X=(0,1) $?

Let $f:X\to X $ be continuous. Show that if $X=[0,1] $, $f $ has a fixed point(i.e. there exists $x$ such that $f (x)=x$). What happens if $X $ equals $[0,1) $ or $(0,1) $? First part of the ...
2
votes
2answers
94 views

Proving $\lim_{h\to0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}=f''(a)$

The following question can be found in Bartle, Introduction to real analysis as well as Walter Rudin, Principle of Mathematical analysis: Let $f$ be differentiable function defined on an open ...
2
votes
1answer
43 views

Vitali Covering?

This is the problem... Let $E$ be a subset of $R$ with $m∗(E) < ∞$, and let $K$ be a collection of compact intervals $I$ covering $E$. Show that there exists a positive constant $β$ and a finite ...