# 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.

93,706 questions
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### Finding Borel sets

Consider a function $f:\mathbb R\to\overline{\mathbb R}$ defined as $$f(x)=\begin{cases}\frac{1}{x}, x\neq 0\\ \infty, x=0 \end{cases}$$ Is $f$ Borel-measurable? I followed the answer given here ...
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### Explanation of Nested Interval Theorem

I have a question relating to the theorem about nested intervals. I understand it until the last expression where it is stated that the interval $[a,b]$ is included in the intersection of those ...
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### Problem Concerning Cauchy Principle for Sequences.

I have a question but can't seem to figure out how to solve it. The problem states: Let's consider a sequence $x_n$, such that $x_n\to a$, as $n \to \infty$. Using the Cauchy Principle prove that (a)...
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### $(x+y)^r \le x^r+y^r$ when $r \in (0,1)$ and $x,y$ are real positive numbers. [duplicate]

I'm sorry for the silly question, but I have a doubt. Given two positive real numbers $x,y$ and taking $0<r<1$, is it true that $$(x+y)^r \le x^r+y^r?$$ In all the examples I considered, ...
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### Is this product strictly positive?

Let $p\in (0,1)$ and $\varepsilon\in (0, 1)$ be fixed. For all $i\in \mathbb N$ we define $p_i=1+(p-1)(1-\varepsilon)^i$. Is it possible to prove that $$\prod_{i=1}^{+\infty}\frac{p_i}{2-p_i}>0$$? ...
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### Proof of Density of Rational Numbers as given in Howie

I'm working out of John M. Howie's Real Analysis (2001). In section 1.4 (Exercise 1.5), we're asked the following: Let $x$ and $y$ be real numbers, with $x < y$. Show that, if $x$ and $y$ are ...