# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

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### Is it true that $0.999999999\dots=1$?

I'm told by smart people that $$0.999999999\dots=1$$ and I believe them, but is there a proof that explains why this is?
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### How to define a bijection between $(0,1)$ and $(0,1]$?

How to define a bijection between $(0,1)$ and $(0,1]$? Or any other open and closed intervals? If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that'...
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### How can you prove that a function has no closed form integral?

I've come across statements in the past along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations: addition/subtraction ...
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### Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$ [closed]

Could any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$? Thank you.
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### Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$

I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using W|A, i may guess that the limit is $\frac{1}{2}$ that is a pretty interesting and nice result. I ...
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### Discontinuous derivative. [duplicate]

Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-...
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### The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$

What is the sum of the 'second half' of the harmonic series? $$\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$ More precisely, what is the limit of the above sequence of partial ...
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### When can a sum and integral be interchanged?

Let's say I have $\int_{0}^{\infty}\sum_{n = 0}^{\infty} f_{n}(x)\, dx$ with $f_{n}(x)$ being continuous functions. When can interchange the integral and summation? Is $f_{n}(x) \geq 0$ for all $x$ ...
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### If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$

Let $1\leq p < \infty$. Suppose that $\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite), $f_k \to f$ almost everywhere, and $\|f_k\|_{L^p} \to \|f\|_{L^p}$. Why is ...
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### Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
60k views

### Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs]

This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as ...
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### Proof of Convergence: Babylonian Method $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$

a) Let $a>0$ and the sequence $x_n$ fulfills $x_1>0$ and $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$ for $n \in \mathbb N$. Show that $x_n \rightarrow \sqrt a$ when $n\rightarrow \infty$. I have ...
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### Set of continuity points of a real function

I have a question about subsets $$A \subseteq \mathbb R$$ for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize ...
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### Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.

Is the following true? Let $x_n$ be a sequence with the following property: Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$. I ...
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By looking through an book, I found this interesting series To prove that: \tan(\theta)+\tan \left(\theta+ \frac{\pi}{n} \right) + \tan\left(\theta + \frac{2\pi}{n}\right) + \dots + \tan \left (\... 6answers 2k views ### How to prove \sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6? I'd like to find out why \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6 \end{align} I tried to rewrite it into a geometric series \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = \sum_{n=... 6answers 21k views ### Does  \int_0^{\infty}\frac{\sin x}{x}dx  have an improper Riemann integral or a Lebesgue integral? In this wikipedia article for improper integrals, \int_0^{\infty}\frac{\sin x}{x}dx $$is given as an example for the integrals that have an improper Riemann integral but do not have a (proper) ... 2answers 725 views ### What are BesselJ functions? I solved an integration on mathematica which gives BesselJ functions and some other terms. I explored mathematica help and google but could not understand the difference between different types of ... 3answers 37k views ### Does convergence in L^{p} implies convergence almost everywhere? If I know \|f_{n}(x) - f(x)\|_{L^{p}(\mathbb{R})} \rightarrow 0 as n \rightarrow \infty, do I know \lim_{n \rightarrow \infty}f_{n}(x) = f(x) for almost every x? 3answers 7k views ### Multiples of an irrational number forming a dense subset Say you picked your favorite irrational number q and looking at S = \{nq: n\in \mathbb{Z} \} in \mathbb{R}, you chopped off everything but the decimal of nq, leaving you with a number in [0,1]... 4answers 7k views ### Can there be two distinct, continuous functions that are equal at all rationals? Akhil showed that the Cardinality of set of real continuous functions is the same as the continuum, using as a step the observation that continuous functions that agree at rational points must agree ... 5answers 6k views ### Nonzero f \in C([0, 1]) for which \int_0^1 f(x)x^n dx = 0 for all n As the title says, I'm wondering if there is a continuous function such that f is nonzero on [0, 1], and for which \int_0^1 f(x)x^n dx = 0 for all n \geq 1. I am trying to solve a problem ... 3answers 3k views ### No continuous function switches \mathbb{Q} and the irrationals Is there a way to prove the following result using connectedness? Result: Let J=\mathbb{R} \setminus \mathbb{Q} denote the set of irrational numbers. There is no continuous map f: \mathbb{R} \... 6answers 45k views ### When can you switch the order of limits? Suppose you have a double sequence \displaystyle a_{nm}. What are sufficient conditions for you to be able to say that \displaystyle \lim_{n\to \infty}\,\lim_{m\to \infty}{a_{nm}} = \lim_{m\to \... 8answers 7k views ### Evaluating \int_0^\infty \sin x^2\, dx with real methods? I have seen the Fresnel integral$$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$evaluated by contour integration and other complex analysis methods, and I have found these methods to be the ... 3answers 10k views ### What is the limit of n \sin (2 \pi \cdot e \cdot n!) as n goes to infinity? I tried and got this$$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$where m is an ... 2answers 2k views ### Show that e^{x+y}=e^xe^y using e^x=\lim_{n\to\infty }\left(1+\frac{x}{n}\right)^n. I was looking for a proof of e^{x+y}=e^xe^y using the fact that$$e^x=\lim_{n\to\infty }\left(1+\frac{x}{n}\right)^n.$$So I have that$$\left(1+\frac{x+y}{n}\right)^n=\sum_{k=0}^n\binom{n}{k}\frac{...
The set of all $\mathbb{R\to R}$ continuous functions is $\mathfrak c$. How to show that? Is there any bijection between $\mathbb R^n$ and the set of continuous functions?