# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, the least upper bound property; and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus. This tag can also be used for more advanced topics, like measure theory.

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### Limits and convolution

Let $f,g \in L^2(\mathbb{R^n})$, $\{ f_n \}, \{ g_m \} \subset C^\infty_0(\mathbb{R}^n)$ (infinitely differentiable functions with compact support) where $f_n \to f$ in $L^2$, and $g_n \to g$ in $L^2$....
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### normed vector spaces

Suppose $A$ is a normed vector space and $a_i \in A$ and we have the following sequence $$\Vert a_1\Vert \leq \Vert a_2 \Vert \leq \cdots \leq \Vert a_n \Vert$$ for any $1\leq i<n-1$. ...
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### the Riemann integrability of inverse function

If $f \colon [a,b] \rightarrow [c,d]$ is a bijection, $f\in \mathcal{R}$ and $f^{-1}$ exists, then prove or disprove that $f^{-1} \in \mathcal{R} [c,d]$. Remark: I tried to use integration by parts ...
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### Why isn't this simpler partition enough?

I'm trying to figure out a lecture example given on our Analysis course. We are currently going through Riemann integrals. Let $g:[0,1] \to R, g(x) = 1$ when $x \in [0, \frac{1}{2}]$ and $g(x) = 2$ ...
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### An application of Fubini's theorem

Let $v_n$ be the volume of the unit ball in $\mathbb{R}^n$. Show by using Fubini's theorem that $$v_n = 2\,v_{n-1}\,\int_0^1{(1-t^2)^{(n-1)/2}\,dt}\,.$$ I've been trying to do this inductively, and ...
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### If $\sum^n_{i=1}{f(a_i)} \leq \sum^n_{i=1}{f(b_i)}$. Then $\sum^n_{i=1}{\int_0^{a_i}{f(t)dt}} \leq \sum^n_{i=1}{\int_0^{b_i}{f(t)dt}}$?

I have been puzzling over this for a few days now. (It's not homework.) Suppose $f$ is a positive, non-decreasing, continuous, integrable function. Suppose there are two finite sequences of positive ...
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### Defining dense subsets of $\mathbb{R}$

Here is a quote from Derbyshire's "Unknown Quantity" The rational numbers are "dense." This means that between any two of them you can always find another one. This being a pop. math book, this is ...
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### between Borel $\sigma$ algebra and Lebesgue $\sigma$ algebra, are there any other $\sigma$ algebra?

Is there any $\sigma$-algebra that is strictly between the Borel $\sigma$-algebra and the Lebesgue $\sigma$-algebra? How about not in between the two, but in general, are there any other $\sigma$ ...
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I'm having an argument about what the notation of $\lim$ means. Assume you have $f_n: X \rightarrow \mathbb{R}$. Are the following two sets equal: $$\{ x \ |\ (f_n(x)) \ \text{converges} \} = ... 3answers 193 views ### Average value of the Fractional Part of a C^{k}-function The mean value of the fractional part \{ x \} on \mathbb{R}_+ is clearly \tfrac{1}{2}. I'd like to know if a similar statement holds for \{ f(x) \} where f \colon X \to \mathbb{R}_{+} is a ... 1answer 122 views ### If p\in\mathbb{Q}^+, for any x\in\mathbb{R}, there is a rational q\in x such that p+q\not\in x I just read about the construction of the real numbers in Enderton's Elements of Set Theory, and am now trying to go through all the exercises. Enderton chooses to construct the reals with left sides ... 2answers 828 views ### Chicken-Egg problem with Fubini’s theorem Fubini's theorem states that if you have a double integral over a function f(x,y), then you can compute the integral as an iterated integral, if f(x,y) is in \mathcal{L}^1. But to find out if f... 3answers 972 views ### If g\geq2 is an integer, then \sum\limits_{n=0}^{\infty} \frac{1}{g^{n^{2}}}  and  \sum\limits_{n=0}^{\infty} \frac{1}{g^{n!}} are irrational How do we show that if g \geq 2 is an integer, then the two series$$\sum\limits_{n=0}^{\infty} \frac{1}{g^{n^{2}}} \quad \ \text{and} \ \sum\limits_{n=0}^{\infty} \frac{1}{g^{n!}} both converge ...
In this question on Math.SE, I asked about Ramanujan's (ridiculously close) approximation for counting the number of 3-smooth integers less than or equal to a given positive integer $N$, namely, \...