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Mathematics Stack Exchange

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.

Real analysis is a branch of mathematical analysis, which deals with the 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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Topology of uniform convergence?

Stone starts with an arbitrary compact Hausdorff space X and considers the algebra C(X,R) of real-valued continuous functions on X, with the topology of uniform convergence I am having a hard time understanding what the topology of uniform…
Zeus
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Cauchy-Schwarz inequality for bilinear forms valued in an abstract vector space

This question is perhaps a little vague; part of what I want to know is what question I should ask. First, recall the following form of the Cauchy-Schwarz inequality: let $V$ be a real vector space, and suppose $(\cdot, \cdot) : V \times V \to…
Nate Eldredge
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Inequality : $\Big(\frac{x^n+1+(\frac{x+1}{2})^n}{x^{n-1}+1+(\frac{x+1}{2})^{n-1}}\Big)^n+\Big(\frac{x+1}{2}\Big)^n\leq x^n+1$

I have the following problem to solve : Let $x,y>0$ and $n>1$ a natural number then we have : $$\Big(\frac{x^n+y^n+(\frac{x+y}{2})^n}{x^{n-1}+y^{n-1}+(\frac{x+y}{2})^{n-1}}\Big)^n+\Big(\frac{x+y}{2}\Big)^n\leq x^n+y^n$$ The problem is equivalent…
Erik Satie
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Approximation of Semicontinuous Functions

Assume that $k \in \mathbb{N}$ and $f : \mathbb{R}^d \rightarrow [0,\infty)$ is lower semicontinuous, i.e. $f(x) \leq \liminf_{y \rightarrow x} f(y)$ for all $x \in \mathbb{R}^d$. Does there exist an increasing sequence of $k$-times continuously…
x_Y_z
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Every complete, countable metric space has a discrete, dense subset.

Given a complete, countable metric space, say $X$, I'd like to show it has a discrete, dense subset. This seems like an application of the Baire Category Theorem, but that doesn't seem to go anywhere. Any help would be appreciated.
Alexander Sibelius
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Bounding a polynomial from below

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \begin{equation} \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \end{equation} where \begin{equation} b_j =…
bgsk
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How did I solve this (triply logarithmic) equation?

In an optimiation problem I came across the following daunting equation: $$ \log \left(\frac{1-t_2}{1-y}\right) \log \left(\frac{(1-x) (t_1-t_2)}{(1-t_1) (x-y)}\right) \log \left(\frac{t_2 x}{t_1 y}\right)\\=\log \left(\frac{t_2}{y}\right) \log…
Thomas Ahle
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If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous

From Pugh's analysis book, prelim problem 57 from Chapter 4: Let $f$ and $f_n$ be functions from $\Bbb R$ to $\Bbb R$. Assume that $f_n(x_n)\to f(x)$ as $n\to\infty$ whenever $x_n\to x$. Prove that $f$ is continuous. (Note: the functions $f_n$ are…
Aden Dong
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Why don't we use closed covers to define compactness of metric space?

I'm a beginner in metric space. So many books I've read, there is only the notion of open covers. I want to know why do we worry about open covers to define the compactness of metric spaces and why don't we use closed covers? What is the problem in…
Jimmy
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A question about converging derivatives

Suppose $f \in C^{\infty}(\mathbb{R})$ and $\forall x \in \mathbb{R} \text{ } \exists \lim_{n \to \infty} f^{(n)}(x) = g(x)$. Does this mean that $$ \exists a \in \mathbb{R} \forall x \in \mathbb{R} \text{ } g(x) = ae^x? $$ If $f^{(n)}$…
Chain Markov
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Holder's inequality for infinite products

In analysis, Holder's inequality says that if we have a sequence $p_1, p_2, \ldots, p_n$ of real numbers in $[1,\infty]$ such that $\sum_{i=1}^n \frac{1}{p_i} = \frac{1}{r}$, and a sequence of measurable functions $f_1, f_2, \ldots, f_n$, then…
JHF
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Methods to solve $\int_{0}^{\infty} \frac{e^{-x^2}}{x^2 + 1}\:dx$

I have a feel this will be a duplicate question. I have had a look around and couldn't find it, so please advise if so. Here I wish to address the definite integral: \begin{equation} I = \int_{0}^{\infty} \frac{e^{-x^2}}{x^2 +…
user150203
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Evaluation of definite Integral

Evaluate $$ \int_{\ln(0.5)}^{\ln(2)}\left( \frac{\displaystyle\sin x \frac{\sqrt{\sin^2(\cos x)+\pi e^{(x^4)}}}{1+(xe^{\cos x}\sin x)^2}+ 2\sin(x^2+2)\arctan\left(\frac{x^3}{3}\right) } {\displaystyle 1+e^{-\frac{x^2}{2}}+x^7 \sin(-\pi …
DeeJay
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How does one begin to even write a proof?

I'm in my first proof based class and I'm just having a lot of trouble writing proofs. I mean I know it's not going to come natural and it will take time, but seroiusly, how does someone begin to write a proof and formulate a game plan? For example,…
TheHopefulActuary
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Finding $\int^1_0 \frac{\log(1+x)}{x}dx$ without series expansion

I was trying to evaluate $$\int^1_0 \frac{\log(1+x)}{x}dx.$$ I expanded $\log(1+x) $ as $x -\frac{x^2}{2}... $ and got the answer. I would like to know if there is any way to do it without series expanding.
Ishan Banerjee
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