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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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Distance of two points in $\mathbb{R}^N$ $\leq$ length of curve

Let $\gamma$ be a $C^1$-curve and $x,y \in \mathbb{R}^N$ with $\gamma: [0,1] \longrightarrow \mathbb{R}^N, \gamma(0)=x, \gamma(1)=y$. My intuition tells me that $||x-y|| \leq \ell(\gamma)$, where $\ell$ is the length of a curve. Is this true and how…
oac
  • 440
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Find all roots of the equation :$(1+\frac{ix}n)^n = (1-\frac{ix}n)^n$

This question is taken from book: Advanced Calculus: An Introduction to Classical Analysis, by Louis Brand. The book is concerned with introductory real analysis. I request to help find the solution. If $n$ is a positive integer, find all roots of…
jiten
  • 4,524
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2 answers

Calculate $\lim _{n\rightarrow +\infty} \sum _{k=0}^{n} \frac{\sqrt{2n^2+kn-k^2}}{n^2}$

Calculate $$\lim _{n\rightarrow +\infty} \sum _{k=0}^{n} \frac{\sqrt{2n^2+kn-k^2}}{n^2}$$ My try: $$\lim _{n\rightarrow +\infty} \sum _{k=0}^{n} \frac{\sqrt{2n^2+kn-k^2}}{n^2}=\lim _{n\rightarrow +\infty} \frac{1}{n} \sum _{k=0}^{n}…
MP3129
  • 3,195
6
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Bounding the absolute value of a function with an integral

I am having trouble with the following problem in analysis: Suppose that $f, f^\prime \in C([0, 1])$. Prove that for all $x \in [0, 1]$ $$ |f(x)| \leq \int_0^1 (|f(t)| + |f^\prime (t)|) dt. $$ Any pointers? I have tried writing this as a Riemann…
onesix
  • 173
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4 answers

Let $\displaystyle f$ be differentiable, $\displaystyle f(x)=0$ for $|x| \geq 10 $ and $g(x)=\sum_{k \in \mathbb Z}f(x+k).$

I came across the following problem that says: Let $\displaystyle f \colon \mathbb R \rightarrow \mathbb R$ be differentiable function and $\displaystyle f(x)=0$ for $|x| \geq 10.$ Let $g(x)=\sum_{k \in \mathbb Z}f(x+k).$ Then which of the…
user52976
6
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2 answers

Show that $\lim((\int_{a}^{b}f^{n})^\frac{1}{n})=\sup\{f(x):x\in[a,b]\}$

Another exercise (this one is 7.2.18) from "Introduction to Real Analysis" by Bartle and Sherbert that I'm struggling with: Let $f$ be continuous on $[a,b]$, let $f(x)>=0$ for $x\in[a,b]$, and let $M_{n}:=(\int_{a}^{b}f^{n})^\frac{1}{n}$. Show that…
Crni
  • 647
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4 answers

For which $a>0$ series is convergent?

For which $a>0$ series $$\sum { \left(2-2 \cos\frac{1}{n} -\frac{1}{n}\cdot \sin\left( \sin\frac{1}{n} \right) \right)^a } $$ $(n \in \mathbb N)$ is convergent? My try:From Taylor theorem I know that:$$a_{n}={ \left(2-2 \cos\frac{1}{n}…
MP3129
  • 3,195
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Is this Riemann-Integrable?

Let $\lfloor x\rfloor$ be the integer part function. Let $$f(x)= \sum_{n=0}^\infty \frac{(-1)^n}{2^n} \lfloor nx\rfloor$$ Is $f(x)$ Riemann-Integrable ? Thanks in advance
thetruth
  • 1,852
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5 answers

Prove the limit problems

I got two problems asking for the proof of the limit: Prove the following limit: $$\sup_{x\ge 0}\ x e^{x^2}\int_x^\infty e^{-t^2} \, dt={1\over 2}.$$ and, Prove the following limit: $$\sup_{x\gt 0}\ x\int_0^\infty {e^{-px}\over {p+1}} \,…
Scorpio19891119
  • 2,345
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3 answers

Function fails to be of bounded variation

Let $f$ fail to be of bounded variation on [0,1]. Show that there is a point $x_0$ in [0,1] such that $f$ fails to be of bounded variation on each nondegenerate closed subinterval of [0,1] that contains $x_0$. I'm trying proving this directly, but I…
Libertron
  • 4,415
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2 answers

Norms on the reals

On the real numbers the absolute value is a norm on this vector space. We can also define the norm of $x$ to be $c|x|$, where $c>0$ is a constant. Are they the only norms on the real numbers? If not, what are other norms on the real numbers?
dxdydz
  • 1,371
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Sequences not in $l^p$

I am wondering if there is an easy sequence $x_n \in \mathbb R$ with $x_n \to 0$ and $x_n \notin l^p$ for all $1 \le p < \infty$. I found $x_n = (\log n)^{-1}$ satisfies $x_n \to 0$ and $x_n \notin l^p$ because $\sum_{n=2}^\infty |x_n|^p \ge…
mathvisitor
  • 65
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1 answer

Open sets and intersections

Suppose $G$ is an open subset of the real number that is not upper bounded. Is there a real number $x > 0$ such that the set of all integer multiples of $x$ intersects $G$ at infinitely many points? That is, is it true that $\exists x \in…
IUissopretty
  • 137
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1 answer

Rudin Theorem 1.21

The Theorem 1.21 at page 10 of Rudin's Book states that For every real $x>0$ and every integer $n>0$ there is one and only one positive real $y$ such that $y^n=x.$ This number $y$ is written $\sqrt[n]{x}$ or $x^{1/n}$. I do not understand the first…
Sangcheol Choi
  • 287
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2 answers

Is the set of all bounded sequences complete?

Let $X$ be the set of all bounded sequences $x=(x_n)$ of real numbers and let $$d(x,y)=\sup{|x_n-y_n|}.$$ I need to show that $X$ is a complete metric space. I need to show that all Cauchy sequences are convergent. I appreciate your help.
Klara
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