Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

Real analysis is a branch of mathematical analysis, which deals with real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness, and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

145439 questions
5
votes
3 answers

If the integral of a series of functions converges to zero, does that series also converge pointwise to zero?

Suppose I have the condition $\int |f_n| \rightarrow 0$. Does this give me the condition that $f_n \rightarrow 0$ pointwise? It seems to be true, except that if you redefine up to countably many points of the $f_n$, it will not change the integral,…
Wyatt Gregory
  • 423
5
votes
3 answers

Definition of Open set.

I'm confused about the definition of Open set "A set Z is open, if for every element z in Z there's a r>0 such that there's an open ball contained with radius r in Z" I read the proof about "An open ball is an open set" What I don't understand is:…
Max Villegas
  • 99
5
votes
1 answer

co-norm of an invertible linear transformation on $\mathbb{R}^n$

$|\;|$ is a norm on $\mathbb{R}^n$. Define the co-norm of the linear transformation $T : \mathbb{R}^n\rightarrow\mathbb{R}^n$ to be $m(T)=inf\left \{ |T(x)| \;\;\;\; s.t.\;|x|=1 \right \}$ Prove that if $T$ is invertible with inverse $S$ then…
Ian
  • 1,391
5
votes
1 answer

How to make open covers disjoint in $\mathbb{R}$?

Say I have some set $E\subset \mathbb{R}$, and an open cover $\cup A_i \supset E$. Is there a general algorithm that makes this cover disjoint?
jack
  • 1,359
5
votes
3 answers

On the distance function $d(f,g) = \sup_{x\in[0,2\pi]}\{|f(x)−g(x)|\}$

Consider the distance function $$ d(f,g) = \sup_{x\in[0,2\pi]}\{|f(x)−g(x)|\}. $$ Let $f_n(x) = \sin(2^nx)$ where $n\in \Bbb N$: show $d(f_n,f_m)≥1$ when $n \neq m$. I know for every $(n,m)$, the function image does show their distance is greater…
Yao Zhao
  • 300
5
votes
1 answer

Unnecessary assumption in Rudin PMA Theorem 5.5

Here is the pertinent section: It seems in the statement of the theorem he has the unnecessary assumption "suppose $f$ is continuous on $[a,b]$". I can't spot where it is used (obviously $f$ is continuous at $x$, but perhaps it's possible that…
Physical Mathematics
  • 4,453
5
votes
2 answers

Ordered field proof that squared terms are zero or positive

I have decided to set up a bounty for one outstanding solution to this problem. The winner with the most clear, confident, solution manual-esque response will be awarded 50 of my participation points. The answer must be clear and unambiguous. You…
user686544
5
votes
2 answers

limit of function and limit of sequence

Let $f: [1,\infty) \rightarrow R$ be a function. Suppose that f is increasing, prove that if $\lim_{x \rightarrow \infty} f(x)$ exists then the sequence $\{f(n)\}_{n=1}^{\infty}$ is convergent. So by definition of limit of function at infinity, let…
Lindsay Duran
  • 1,255
5
votes
5 answers

Convergence of $\sum_{n=1}^\infty \dfrac{n\log n}{e^n}?$

How to test the convergence of $\sum_{n=1}^\infty \dfrac{n\log n}{e^n}?$
Sriti Mallick
  • 6,137
  • 3
  • 30
  • 64
5
votes
1 answer

For which $a\in \mathbb R$ the integral is convergent?

For which $a\in \mathbb R$ the integral is convergent? $$ \int_0^{+\infty} x^{-5a} \ln(1+x^{2a})dx$$ Firstly I tried to use: $$f(x)=\ln (1+x^{2a}), f'(x)=\frac{1}{1+x^{2a}}$$ $$g(x)=\begin{cases}{\frac{1}{1-5a} x^{1-5a} , a\neq \frac{1}{5}\\\ln…
MP3129
  • 3,195
5
votes
3 answers

Is this equation correct? And if so, is this famous?

$$\int_1^{\infty}x^n e^{-x}dx=\frac{1}{e} \sum_{k=0}^n\frac{n!}{(n-k)!}$$ My proof.1 \begin{align*} &\int_{1}^{\infty} e^{-\alpha x} \, \mathrm{d}x = \frac{e^{-\alpha}}{\alpha}, \\ &\Rightarrow \qquad \int_{1}^{\infty} (-x)^n e^{-\alpha x} \,…
Starlight Road
  • 51
5
votes
2 answers

Mean Value Theorem and Intermediate Value Theorem

Let $a,b \in \mathbb{R}$, $a
user52932
  • 403
5
votes
2 answers

Asymptotic inverses of asymptotic functions

Under what conditions are the inverses of asymptotic functions themselves asymptotic? (A simple example where this is not the case is the pair of functions $\log (x)$ and $\log (2x)$.)
Palafox
  • 261
5
votes
2 answers

Equicontinuity of the closure of an equicontinuous set

If E Is equicontinuous in C(X,R), I need to show that $\bar{E}$ is equicontinuous as well. Now $\forall f \in \bar{E}, \exists f_n\in E$ s.t. $f_n\rightarrow f$, thus $\forall \epsilon > 0, \exists n_o \text{ s.t. if } n\geq n_0, |f_n(x)-f(x)|<…
Klara
  • 1,233
5
votes
1 answer

sum of rational and irrational numbers with dedekind cuts

Can someone please help me to prove what is the sum of two rational numbers (is rational obviously but why) with dedekind cuts, what is the sum of rational and an irrational and the sum of two irrational numbers also with the definition of…
art
  • 73
1 2 3
…
99 100