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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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Covering of a compact set

Let $K$ be a compact subset of $\mathbb{R}^n$. Fix a constant $r>0$, I'm wondering whether there exists a finite collection of points $x_1,\dots,x_k \in K$ such that the collection of open balls $\{B(x_i,2r)\}_{i=1}^{k}$ forms an open cover of $K$…
sz3
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If function is differentiable at a point, is it continuous in a neighborhood?

I was reading a proof for the multi-variable chain rule and in the proof the mean-value theorem was used. The use of the theorem requires that a function is continuous between two points. Hence the motivation for the question, if a function is…
Tony
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Prove or disprove: If $\sum a_n$ converges conditionally, then $\sum n^2 a_n$ diverges

Intuitively my guess is that the statement is true but i've struggled to find a way to show it rigorously. The reason I believe it to be true is since $\sum n^2 a_n$ converging seems like a "strong" condition and for some obvious candidates that…
ChairmanMeow
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limit with $\arctan$

I have to find the limit and want ask about a hint: $$\lim_{n \to \infty} n^{\frac{3}{2}}[\arctan((n+1)^{\frac{1}{2}})- \arctan(n^{\frac{1}{2}})]$$ I dont have idea what to do. Derivatives and L'Hôpital's rule are so hard
aiki93
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Prove if $g$ has a fixed point in $(0,1)$, then $g^{\prime}(1) > 1$.

Let $g : [0, 1] \rightarrow \mathbb{R}$ be twice differentiable with $g^{\prime \prime}(x) > 0$ for all $x \in [0,1]$. Suppose that $g(0) > 0$ and $g(1) = 1$. Prove if $g$ has a fixed point in $(0,1)$, then $g^{\prime}(1) > 1$. My attempt: Define a…
Idonknow
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Prove that $f^{(n)}(x) = 0$ for some $x$.

Suppose that $f : \mathbb{R} \rightarrow \mathbb{R}$ is $n$-times differentiable and $f$ has $n+1$ distinct zeros. Prove that $f^{(n)}(x) = 0$ for some $x$. My attempt: Prove by induction For $n=1$, we have $f$ is differentiable and has $2$ distinct…
Idonknow
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Definition 4.1 Principles of Mathematical Analysis

Definition 4.1 Let X and Y be metric spaces; suppose E $\subset$ X, if $f$ maps E into Y and $p$ is a limit point of E. We write $f(x)$ $\to$ $q$ as $x$ $\to$ p if there is a point $q$ $\in$Y with the following property: $\forall \epsilon>0,…
Brown
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Show that $\lim_{n\to\infty}\frac{\lfloor nx \rfloor}{n}=x$

How do I prove that $\lim_{n\to\infty}\frac{\lfloor nx \rfloor}{n}=x$ for $x\in\mathbb{R}$? I see that $\lfloor nx\rfloor = n\lfloor x \rfloor + \lfloor n(x-\lfloor x \rfloor )\rfloor + O(1)$ but I'm not sure how to deal with the middle term.
Dellli
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Is $\{n!\alpha\},n \in\mathbb{N}$ dense in $R$ ? where $\alpha$ is irrational

Is $\{n!\alpha\},n \in\mathbb{N}$ dense in $[0,1]$ ? where $\alpha$ is irrational. I know that $\{n\alpha\}$ is dense in $[0,1]$? I wanted to generalise it.So, first I thought about $\{n!\alpha\},n \in\mathbb{N}$, but couldn't make any reasonable…
user694028
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1 answer

Derivative of sign function $\operatorname{sgn}(x)$ (in distribution sense).

In the book of Schilling and Partzsch : Brownian motion (in the part of the Tanaka formula), they say that the derivative of $f(x)=\text{sgn}(x)$ is given by $f'(x)=\delta _0(x)$ (in distribution sense). But I find $f'(x)=2\delta _0(x)$ and I don't…
user657324
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1 answer

Find the limit of sequence $a_{n+1} -a_{n}$

Consider the sequence $a_{n}=\sqrt{n}+\sqrt[3]{n}+\cdots+ \sqrt[n]{n}.$ Find the limit of sequence $a_{n+1} -a_{n}.$ One of my attempts is to use the reciprocal of the Stolz-Cesaro lemma, which is true by imposing additional conditions. What is…
medicu
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Pointwise version of Fejer's Theorem

In Principles of Mathematical Analysis Rudin asks that if $f$ is integrable in $[-\pi,\pi]$ of period $2\pi$ and $f(x+)$ and $f(x-)$ exist for some $x$, then $$\lim_{N\rightarrow \infty}\sigma_N(f;x)=\frac{1}{2}[f(x+)+f(x-)].$$ How is this done?
user65375
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Unclear ideas in the proof of the Archimedean Principle

Here is how the principle is laid out in the text: Given real numbers $a$ and $b$, with $a \gt 0$, there is an integer $n \in \mathbb{N}$ such that $b \lt na$ First off what is important about finding an $n$ that satisfies $b \lt na$ It talks…
K. Gibson
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Baby Rudin chapter 2 exercise 8

Exercise 2.8: Is every point of every open set $E\subset R^2$ a limit point of $E$ ? My Solution: Every point of every open set $E\subset R^2$ is a limit point of $E$. [Notation: $N_r(p)$ is the set of all point x such that $0< d(x,p)< r $ ] Since…
Xianglin Zhao
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Continuity of Monotone Functions

Let f be a monotone function on the open interval (a,b). Then f is continuous except possibly at a countable number of points in (a,b). Assume f is increasing. Furthermore, assume (a,b) is bounded and f is increasing on the closed interval [a,b].…
user58289
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