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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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Real analysis: $\epsilon-N$ proof

Here's my proof for the fact that the sequence $\frac{n+6}{n^2-7}$ converges to $0$. Can someone verify it? Let $\epsilon > 0$. Pick N = $max\{2, \frac{7}{\epsilon}\}$. Then, if $n > N$, $\left|\frac{n+6}{n^2-6}\right| = \frac{n+6}{n^2-6}$, since…
Ryan Stiles
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About compact metric space

Let $(E,d)$ a metric space. Supose that exists $r>0$ such that $\overline{B(x,r)}$ is compact for each $x\in E$. Let $A\subset E$ compact. Prove that the set $$ \lbrace x\in E: d(x,A)\leq s\rbrace$$ is compact for each $s
El Peluca Sapeeee
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proving that some series converges uniformly

How can I prove that the series $ \sum\limits_{n = 1}^\infty {\frac{{\sin \left( {nx} \right)}} {n}} $ converges uniformly on the interval $ [\varepsilon ,2\pi - \varepsilon ]\,\,\varepsilon > 0 $ In general , it´s difficult to me to prove…
August
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Discuss the continuity of the given function

Let $f(x)$ be the function defined on the interval $(0,1)$ by $f(x) = \left\{ \begin{array}{l l} x& \quad \text{if $x$ is rational}\\ 1-x& \quad \text{otherwise} \end{array} \right.$ Discuss the continuity of $f(x)$ in the given interval. Please…
idpd15
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If a sequence {$f_n$} pointwise convergent on a finite set $D$ which is a subset of $\mathbb R$.Then can we say that the convergence is uniform?

If a sequence {$f_n$} pointwise convergent on a finite set $D$ which is a subset of $\mathbb R$.Then can we say that the convergence is uniform? If the answer is yes then I have some problem here: Let us suppose that $D$={$0$,$1$} and $f_n(x)$=$x^n$…
liesel
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Proving $\ln e = 1$

Using the definition $$ \ln x = \int_1^x \frac{dt}{t}, $$ is it possible to show that $\ln e = 1$ without showing first that $\exp$ and $\ln$ are inverse functions? Here, $e$ is defined by the series $$ e = \sum_{k=0}^\infty \frac{1}{k!}. $$ EDIT:…
David Zhang
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Easy example to illustrate that the following normed space is not complete

Consider the vector space $C([0,1])$ of real-valued continuous functions on $[0,1]$ endowed with the standard norm: $$ \Vert f\Vert_2 = \sqrt{\int_0^1 f(x)^2 dx}.$$ I know that this normed space is not complete. Is this because the function $f_(x)…
Bana
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Prove that there is $\{x_n\}_{n=1}^{\infty}$ such that $ \sum_{n=1}^{\infty}f(x_n)=\infty$

I have to prove the following statement, but I am not sure that my solutions is correct: Let $f:[0,1] \rightarrow\mathbb{R}$ be a function such that $f(x)>0$ for all $x\in[0,1]$, but which is otherwise arbitrary. Prove that there is a sequence …
Lorenzo Stanca
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Show the set of the limits of all the subsequence of a bounded set contains sup and inf.

Let $(X_n)$ be a bounded sequence, and let $E$ be the set of subsequential limits of $(X_n)$. Prove that $E$ is bounded and contains $\sup E$ and $\inf E$. Does this ask us to prove limsup and liminf exists? Could you help me ?
ZHJ
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Prove that $\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k = 1}^{n}{f(\frac{k}{n}) }$ $=\int_0^1 f(x)dx.$

Question: Let $f$ be continuous on $[0,1]$. Prove that $\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k = 1}^{n}{f(\frac{k}{n}) }$ $=\int_0^1 f(x)dx.$ where $k=0,1,...,n.$ Attempt: I don't even know where to start. It makes sense reading the sum, as…
sillyme
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proof that reals are uncountable

I found this proof in Goldberg's Methods of Real Analysis: Assume the $\mathbb{R} = \{x_1, x_2, ... \}$ are countable. Let $I_1$ be the interval $(x_1-1/4, x_1+1/4)$, $I_2$ be the interval $(x_2-1/8,x_2+1/8)$, $I_n$ be the interval…
littlejim
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How to solve double limit $\lim_{m\to\infty}[\lim_{n\to\infty}(\cos(m!\cdot \pi\cdot x))^{2n}]$

Please provide some hints as to how to solve questions with double limits such as this: $$\lim_{m\to\infty}\left[\lim_{n\to\infty}(\cos(m!\cdot \pi\cdot x))^{2n}\right]$$ One of the things I did was convert the original function…
user130113
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Problem about uniform continuity on $[0,\infty)$

Can you help me to prove this statement? I found it in this book…
greyls
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Uniform convergence on union of intervals

Let $a
kris91
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the Taylor series expansion about the origin in the region $|x| < 1$ for the given function

Write down the Taylor series expansion about the origin in the region $|x| < 1$ for the function $$f(x)=x\tan^{-1}x-\frac{1}{2}\log(1+x^2)$$ I cant find the general formula for $f^{(n)}(x)$.an I get some help?
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