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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
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"Counterexample" to L'Hopital's Rule as an exercise in "Understanding Analysis" by Stephen Abbott, 2nd Edition

The following is taken from the 2nd Edition of Stephen Abbott's book "Understanding Analysis". I must admit that I am a huge fan of this book. Theorem 5.3.6 (L'Hopital's Rule, 0/0 case) Let $f$ and $g$ be continuous on an interval containing $a$,…
Simon
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Dense subset of the Cantor set

Prove that the set of endpoints of removed intervals in the Cantor middle thirds set is a dense subset of the Cantor set. Attempt at proof: Since each subinterval is of length $(1/3)^n$, any point contained in $K_n$ is at a distance of less than…
Heisenberg
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Is it obvious that $q = \frac{2p+2}{p+2} > p$, how does one easily show $q > p$?

I was going through walter Rudin's first example when trying to show $A = \{p \in Q_+ : 0
Charlie Parker
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Cauchy Sequences in R Converge: to What?

I am trying to show the completeness of R, using the LUB property. Problem is that I don't know, given a Cauchy sequence,where the limit would come from; I can check if a sequence {an} converges to a specific value, but I don't know how to come up…
user6600
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moving limit through the integral sign

Let $F(\alpha) = \int\limits_0^{\pi/2} \ln( \alpha^2 - \sin^2 x) \mathrm{d} x $ where $\alpha>1$ Im tempting to argue that $F(1) = \int\limits_0^{\pi/2} \ln (1 - \sin^2 x) dx = \int\limits_0^{\pi/2} \ln \cos^2 x dx $ But, $\alpha > 1$. Thus, the…
user139708
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Write $(0,1)$ as a countable union of disjoint open intervals

I saw a proof of the following theorem. Every open subset $\mathcal{O}$ of $\mathbb{R}$ can be written uniquely as a countable union of disjoint open intervals. The proof was convincing, but can anyone help me writing out explicitly such a…
user34183
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On the Gevrey class of bump functions.

A Gevrey function of order $\rho \geq 1$ is a smooth function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ that satisfies the estimate $$|\partial^\alpha f|\leq AL^{|\alpha|}\alpha!^\rho $$ for some positive constants $A,L$, where $\alpha$ is an arbitrary…
goonfiend
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How to show that $\sqrt[3]{26+15\sqrt{3}}+\sqrt[3]{26-15\sqrt{3}} \in \mathbb{Z}$?

How to show that the following is true? $$\sqrt[3]{26+15\sqrt{3}}+\sqrt[3]{26-15\sqrt{3}} \in \mathbb{Z}$$ I have tried to set $$\sqrt[3]{26+15\sqrt{3}}+\sqrt[3]{26-15\sqrt{3}} = r,$$ $$a=\sqrt[3]{26+15\sqrt{3}},$$ $$b=\sqrt[3]{26-15\sqrt{3}},$$ and…
user50224
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Difference in Sigma additivity and finite additivity

A set function $\mu(A)$ is called a measure if 1. ... 2 ... 3. $\mu$ is additive in the sense that if $A$ is a set in $\mathscr{S}_{\mu}$ such that $$A = \bigcup_{k=1}^{n} A_{k}, $$ where $A_{1}, \cdots , A_{n}$ are pairwise disjoni sets in…
Taln
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If $f \colon [-1,1] \to \mathbb R$ satisfies $\vert f'' \vert \le k$ then $\vert f \vert \le \frac{k}{2}$

Let $f \colon [-1,1] \to \mathbb R$ be a twice differentiable function s.t. $f(-1)=f(1)=0$ and there exists $k>0$ s.t. $\vert f''(x) \vert \le k$ for every $x \in [-1,1]$. Show that $$ \max_{[-1,1]}\vert f \vert \le \frac{k}{2}. $$ The book…
Romeo
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Nonlinear functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ that preserve or grow the angle between any two vectors?

Do there exist differentiable almost-everywhere functions on $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\frac{|\langle x, y \rangle|}{|x||y|} \geq \frac{|\langle f(x), f(y) \rangle|}{|f(x)||f(y)|}$? How does one go about constructing one?
David Warde-Farley
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Theorem 3.17 in Rudin's Analysis

This question refers to Rudin's "Principles of Mathematical Analysis", Theorem 3.17, p.56. In particular let $\left\{s_n\right\}$ be a real sequence and let $E$ be the set of all sub-sequential limits with possibly plus and minus infinity included.…
Manos
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If $\lim_{x\to \infty }f'(x)=0$ then $\lim_{x\to \infty }f(x)$ exist.

I want to show that if $f:\mathbb R\longrightarrow \mathbb R$ (a derivable function) is bounded and s.t. $$\lim_{x\to \infty }f'(x)=0,$$ then $f$ has a limit in $+\infty $. I tried as follow: If $f$ doesn't reach his supremum (let denote it…
MSE
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The limit of periodic function sequence is periodic

Suppose $f_n(x)$ is continuous and periodic, $f_n(x+T_n)=f_n(x)$. If we know that $f_n(x) \to f(x)$ uniformly and $T_n \to T$, can we conclude that $f(x+T)=f(x)$ ?
Takanashi
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How to show that this identity holds?

By comparing the first terms of the Taylor expansion at $0$, it seems that for $|x|<4/27$ the following identity holds: $$\ln\left(\sum_{n=1}^{\infty} \binom{3n}{n}\frac{x^{n-1}}{2n+1}\right)= \sum_{n=1}^{\infty} \binom{3n}{n}\frac{x^n}{n}.$$ I…
Robert Z
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