# 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
88 views
+50

### 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 ...
115 views
+50

10 views

### 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 ...
41 views

26 views

15 views

### 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)...
37 views

### $(x+y)^r \le x^r+y^r$ 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, ...
30 views

62 views

### 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 $$\prod_{i=1}^{+\infty}\frac{p_i}{2-p_i}>0$$? ...
I have a $n$th-order non-linear partial differential in $m$-real variables $x_1,x_2, \ldots, x_m$. Assume a function $f$ satisfies this differential equation. I denote this by $$D f(x_1, x_2, \ldots, ... 0answers 11 views ### \overline{lim} the set of cluster points I am currently reading an article where the author has the following statement. " \overline{\lim}_{t\to\infty} u_t is the set of cluster points of the sequence u_k \subset U which is nonempty ... 1answer 17 views ### Is induction the correct approach here? Let a_1, a_2, \dots, a_n (real numbers) be such that$$a_1 - a_2/3 + \dots + (-1)^{n - 1}a_n/(2n - 1) = 0.$$Prove that$$f(x):= a_1\cos(x) + a_2\cos(3x) + \dots + a_n\cos([2n - 1]x) = 0, for ...
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 ...