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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. .

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Stuck at Evaluating the Riemann-Stieltjes Integral

Let $$\alpha(x) = \left\lbrace \begin{array}{cc} 0 & x=0\\ \dfrac{1}{2^n} & \dfrac{1}{3^n} < x \leq \dfrac{1}{3^{n-1}}\quad n=1,2,...\end{array}\right.$$ Evaluate $$\int_{0}^{1}{x\mathrm{d}\alpha(x)}$$ Attempt: I dont know how to put this formally,…
AAP
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The union of a sequence of infinite, countable sets is countable.

While reading Walter Rudin's Principles of Mathematical Analysis, I ran into the following theorem and proof: Theorem 2.12. Let $\left\{E_n\right\}$, $n=1,2,\dots$, be a sequence of countable sets, and put $$ S=\bigcup_{n=1}^\infty E_n. $$ Then $S$…
wjmolina
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If $f\in L^1(\mathbb{R}^d)$ is uniformly continuous, do its integrals over spheres of increasing radius decay to zero?

To be more precise, if $f\in L^1(\mathbb{R}^d), d>1$ and is uniformly continuous, is it true that $$\lim_{R\to\infty} \int_{|x|=R} |f(x)| \ dS(x) = 0 \ ?$$ where $dS$ is the surface measure on the sphere of radius $R$. If $f\in L^1(\mathbb{R})$ and…
kieransquared
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If A and B are disjoint open sets, prove that they are separated

I just proved this statement like the below. Is this valid or solid proof? Thank you!
InfimumMaximum
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Conjecture: $\lim_{x\to 0^+}f(2x)\log f(x)=0$.

I encountered the following problem: Suppose $f$ is a strictly increasing $C^1$ function defined on $[0,a]$ for some $a>0$, such that $f(0)=f'(0)=0$. Is it true that $\lim_{x\to 0^+}f(2x)\log f(x)=0$? I tried to use the fact that $z^{\alpha}\log…
MrDR
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Motivated by Poisson distributions, prove that: If $n p_n \to \lambda$ then $ \left( 1 - p_n \right)^n \to e^{-\lambda}$

I have this ugly proof that some sequence converges to something and I really don't like it because it seems too hard for no reason... can anyone help me make this more simple? Here it is : We are given a sequence $p_n$ with $n p_n \to \lambda > 0$.…
Patrick Da Silva
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Proving $f$ is not surjective

Given the function $f: \mathbb R \mapsto \mathbb R, f(x) = 2e^x + 3x^2 - 2x + 5,$ I am asked to show that it is not surjective. My book goes about it like this: $f(x) = 2e^x + x^2 - 2x + 1 + 2x^2 + 4 = f(x) = 2e^x + (x-1)^2 + 2x^2 + 4$ $f(x) > 0…
Paul Manta
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How to show that for any real numbers $x,y,z$ $|x|+|y|+|z|\le|x+y-z|+|y+z-x|+|z+x-y|?$

How to show that for any real numbers $x,y,z$$$|x|+|y|+|z|\le|x+y-z|+|y+z-x|+|z+x-y|?$$ I'm don't know how to split RHS.
Sriti Mallick
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Show sequence ${a_n} = \sqrt[n]{{{3^n} + {5^n}}}$ is monotone decreasing

(a) Show that sequence ${a_n} = \sqrt[n]{{{3^n} + {5^n}}}$ is monotone decreasing Proof Let ${a_n} = \sqrt[n]{{{3^n} + {5^n}}} = 5{\left[ {{{\left( {\frac{3}{5}} \right)}^n} + 1} \right]^{\frac{1}{n}}}$ then ${a_k} = 5{\left[ {{{\left(…
Doretomtom
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Distance of scaled point on the unit sphere to an integer

Let $S:=\{ x\in \mathbb{R}^d:||x||_2=1\}$ be the d-dimensional unit sphere, where $||x||_2$ is the euclidean norm. Let $\epsilon>0$ and $s\in S$ be an arbitrary point on the sphere. Is it correct that there exists an $\alpha>0$ and a $k\in…
HyyFly
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Counterexample on mixed partials

I was thinking about Young's and Schwarz's theorem on when do partial derivatives be equal and I was wondering about how smooth can a function whose mixed partial are not equal be. I was wondering there is a function $f:\mathbb{R}^2 \to \mathbb{R}$,…
Jorge
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$||f-f_n||_{L^1} \rightarrow 0$ but $f_n \rightarrow f$ for no $x$

Show that there are $f\in L^1(\mathbb{R}^d)$ and a sequence $\{ {f_n}\}$ with ${f_n}\in L^1(\mathbb{R}^d)$ such that $||f-f_n||_{L^1} \rightarrow 0$ but $f_n \rightarrow f$ for no $x$ Thanks.
catch22
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Hölder 1/2 condition for curve

Does there exist a continuous map $\gamma:[0,1] \to \mathbb{R}^n$ s.t. $$|\gamma(x)-\gamma(y)| \ge |x-y|^{\frac 1 2} $$ for all $x,y \in [0,1]$?
user64494
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Archimedean Proof?

I've been struggling with a concept concerning the Archimedean property proof. That is showing my contradiction that For all $x$ in the reals, there exists $n$ in the naturals such that $n>x$. Okay so we assume that the naturals is bounded above…
Ben Anderson
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