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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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When is the $n$th term sufficient to guarantee convergence of a series

If $K$ is a field complete with respect to a non-archimedean absolute value, then the $n$th term test (checking whether the $n$th term of a series goes to zero) is sufficient to check convergence of a series in $K$. I was wondering if we have any…
Derek Scavo
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Lipschitz-like function which is nowhere differentiable function

I've been trying this problem from Stein, but with no luck. Consider the function $$f_{1}(x)=\sum_{n=0}^{\infty}{2^{-n} e^{2\pi i 2^{n} x} }.$$ a) Prove that $f_{1}$ satisfies $|f_{1}(x)-f_{1}(y)| \leq A_{\alpha}|x-y|^{\alpha}$ for each $\alpha \in…
Anna
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Necessity of the Completeness Axiom in Calculus

A friend asked me today why we need to bother with the Completeness Axiom in calculus. Even though I have taken a real analysis course quite a while ago, I could not answer him and I realized I had the same question. The Completeness "Axiom" for…
TSJ
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Error in my proof?

What is wrong in this proof. It seems correct to me but still doesn't make proper sense. $$\sqrt{\cdots\sqrt{\sqrt{\sqrt{5}}}}=5^{1/\infty}=5^0=1$$ EDIT So does this mean that $5^{1/\infty} = 1$ $(5^{1/\infty})^\infty = 1^\infty$ But according to…
Arulx Z
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Proving $\lim_{x\to 0}f(x)=\infty$.

Suppose we have a function $f:[0,\infty)\to \mathbb{R}$ such that for every $N\in\Bbb{N}$ and every sequence of $\delta_n>0$ such that $\lim_{n\to\infty}\delta_n=0$, there exists $n$ for which $f(\delta_n)\geq N$. Does that imply that $$\lim_{x\to…
user232358
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Real analysis question

I am studing some old exams and came across this one that has me stumped. Suppose $f:[0,1] \rightarrow \mathbb{R}$ is absolutely continuous. Show $$\lim_{n \rightarrow \infty} \sum_{k=0}^{2^{n}-1} \left| f( k/2^{n}+1/2^{n} )-f( k/2^{n}…
Mykie
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For all $\epsilon>0$ there exists $f,g$ such that $\|f\ast g\|_p>(1-\epsilon)\|f\|_1\|g\|_p$.

Let $f\in L^1(\mathbb{R})$ and $g\in L^p(\mathbb{R})$, for $1\leq p\leq\infty$. A well-known is result, called Young's Inequality is that $$\|f\ast g\|_p\leq\|f\|_1\|g\|_p,$$ where $$(f\ast g)(x)=\int_{-\infty}^\infty f(x-y)g(y)\,dy$$ is the…
user228374
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Does the converse of Lusin's theorem hold?

Lusin's theorem says that if $f:[a,b]\to\mathbb{C}$ is a measurable function, then for any given $\varepsilon>0$ there is a continuous function such that $\mu(\{x\in[a,b]:f(x)\neq g(x)\})<\varepsilon$. But I wonder if the converse is true, that is,…
Maclaurin
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exchange max and limit

Let $\lim_{n \rightarrow \infty} a_n = a$ and $\lim_{n \rightarrow \infty} b_n = b$ exist, then is it true that $\lim_{n \rightarrow \infty} \max \{ a_n, b_n \} = \max \{ a, b \}$? I couldn't find this on wiki, but it seems correct. Here's my proof,…
simonzack
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What are all of the connected subsets of $\mathbb{Q}$?

The answer are the singletons of $\mathbb{Q}$. I can show that the open intervals of $\mathbb{Q}$ are disconnected by choosing some irrational in the open set and using it to form a separation. But strangely enough, I am having a hard time seeing…
Peter
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Pointwise Convergence on an Infinite Intersection of Function Images

This is the most interesting real analysis question I have run into thus far (that I understand): For all $n\in\mathbb{N}$, let $f_n(x)$ and $f(x)$ be one-to-one continuous functions such that $$B=\bigcap_{n=1}^{\infty}f_n(A)$$ is a nonempty…
wjmolina
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Calculating the Lebesgue and Riemann integrals

Given $$f(x) = \begin{cases} \frac{1}{q} & \text{if}~~ x = \frac{p}{q},(p,q)=1 ~,p\leqslant q, \text{is rational},~ 0\leqslant x \leqslant 1 \\ 0 & \text{if}~~ x ~~\text{is irrrational}, 0\leqslant x \leqslant 1 \end{cases} $$ I want to…
Joe
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Need help understanding a proof on the least upper bound property of the reals.

I'm trying to learn from a book called "Vector Calculus, Linear Algebra, and differential forms". In chapter 0.5, there's a proof on the least upper bound property of reals. I'm learning the material by myself, so I don't have anyone to ask about…
user269334
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$f(x)$ is non-negative and that $ \int_{-a}^b xf(x) dx = 0 $. Show $ \int_{-a}^bx^2f(x)dx \leq ab\int_{-a}^b f(x)dx $

$a, b > 0$. $f(x)$ is non-negative and integrable on $[-a, b]$ and that $ \int_{-a}^b xf(x) dx = 0 $. Prove that $$ \int_{-a}^bx^2f(x)dx \leq ab\int_{-a}^b f(x) dx $$ Thanks!
leafpile
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Find a subset of the irrationals that is homeomorphic to the irrationals

My Real Analysis professor and I have been trying to construct a particular example in the irrationals but to no avail. The criteria are as follows: Let $\mathbb{J}$ be the set of irrationals and let $\mathbb{J}$ be given the topology inherited from…
Russle
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