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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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A question about Baby Rudin Theorem 2.27 (a)

Theorem 2.27: If $X$ is a metric space and $E \subset X$, then $\bar E$ (the closure of $E$) is closed. The proof says: If $p \in X$ and $p \not \in \bar E$ then $p$ is neither a point of $E$ nor a limit point of $E$. Hence $p$ has a neighborhood…
David Tan
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Pointwise limit of integrable function

Question: Show that the pointwise limit of integrable functions is not necessarily integrable. I am stuck on this question. Here is what I know. Let $(f_n)^{\infty}_{n=1}$ be a series of integrable functions, and let $$\lim_{n \to \infty}…
greg_mankob
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Continuous $f:[0,1]\to\mathbb{R}$ such that $f(0)=f(1)$ and $\forall\alpha\in(0,1)\exists c\in[0,1-\alpha]|f(c)=f(c+\alpha)$?

Let $f:[0,1]\to\mathbb{R}$ continuous such that $f(0)=f(1)$. Is it true that $\forall\alpha\in(0,1)\exists c\in[0,1-\alpha]|f(c)=f(c+\alpha)$? At first I tried to find a counterexample but my intuition says it's true. Then I've got the idea of…
Javi
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If $\alpha$ is an irrational real number, why is $\alpha\mathbb{Z}+\mathbb{Z}$ dense in $\mathbb{R}$?

This is chapter $4$ exercise $25.b$ of Walter Rudin's Principles of Mathematical Analysis, this problem has occupied my mind for a long time, and I haven't been able to solve it, I would like to see an answer to this question. Thanks.
Camilo Arosemena-Serrato
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Does $\lim_{n \to \infty}a_n^{1/n} = 1$ imply $\lim_{n \to \infty} \frac{a_{n + 1}}{a_n} = 1$?

Supose I have a sequence $\{a_n\}$ of positive real numbers such that $\lim\limits_{n \to \infty}a_n^{1/n} = 1$. Is it true that $\lim\limits_{n \to \infty} \frac{a_{n + 1}}{a_n} = 1$ or depends of the sequence that a choose?
LAU
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For every real number $a$ there exists a sequence $r_n$ of rational numbers such that $r_n$ approaches $a$.

How to prove that for every real number $a$ there exists a sequence $r_n$ of rational numbers such that $r_n \rightarrow a$.
JuanSancen
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What's the component interval?

In Apostol, page $51$, he defines what he calls the component interval. I can't find any reference to it on the web. I have some problems with the definition: Let $S \subseteq \mathbb{R}$. An open interval $I$ of $S$ is a component interval if…
nabil
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Existence of a continuous function with pre-image of each point uncountable

Does there exist a continuous function $f : [0, 1] → [0, 1]$ such that the pre-image $f^{−1}(y)$ of any point $y \in [0, 1]$ is uncountable?
Junyu
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If $\sum_{n=1}^{\infty} a_{n}^{3}$ converges does $\sum_{n=1}^{\infty} \frac{a_{n}}{n}$ converge?

Suppose $a_{n}>0$ and the following series converges $\sum_{n=1}^{\infty} a_{n}^{3}$ Does this imply that $\sum_{n=1}^{\infty} \frac{a_{n}}{n}$ converges? I was able to prove that the second series also converges by using the limit comparision…
Mykie
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If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$.

Possible Duplicates: Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$ Slight generalization of an exercise in (blue) Rudin What can we say about $f$ if $\int_0^1 f(x)p(x)dx=0$ for all polynomials $p$? I found a nice…
Potato
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Hausdorff dimension of Cantor set

I know this is probably a easy question, but some steps in the proofs I found almost everywhere contained some parts or assumptions which I think may not be that trivial, so I would like to make it rigorous and clear enough. Here is the…
Sun
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Equivalent definition of absolutely continuous

A function $f$ is absolutely continuous on $[a,b]$ is defined by: for each $\varepsilon>0$, there is a $\delta>0$, for each finite disjoint open interval $\{(c_k,d_k)\}_{k=1}^n$ contained in $[a,b]$, we have $$ \text{if}\,\, \sum_{k=1}^n…
hxhxhx88
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Can infinity be a supremum? Can it be a maximum?

If you consider all the real numbers, is infinity the supremum? What about the maximum? I know the supremum does not have to be in the set and the maximum does, but I'm confused as to how to answer these questions. Are the real numbers still…
user268486
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Is $p\mapsto \|f\|_p$ continuous?

Suppose that $\|f\|_p < \infty$ for all $1\leq p < p'$, I want to know if the the following is true and in that case how to show it $p \mapsto \|f\|_p$ is continuous on $[1,p')$ Or maybe we need to impose some more constraints such as finite…
Hawii
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Difference between boundary point & limit point.

A limit point is just a accumulation point whose neighbourhood contains infinitely many elements of the sequence. Is there any difference between boundary point & limit point? I've read in another question here that all boudary points are limit…
user142971
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