# Real analysis Definition and 106 Discussions

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions.

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1. ### Analysis Study plan for Functional Analysis - Recommendations and critique

Hello, PF! It’s been a while since I last posted. I am looking for a critique and recommendations regarding my study plan towards Functional Analysis and applications (convex optimization, optimal control), but first, some background: - This plan is in preparation for my master’s thesis, I...
2. ### Proving that there is a sequence in S, such that ##\lim s_n = \sup S##

Let ##S=\{s_n:n∈N\}##. ##\sup S## is the least upper bound of S. For any ϵ>0, we have an m such that ##\sup S−\epsilon \lt s_m## ##\sup S−s_m \lt \varepsilon## ##|\sup S−s_m| \lt \varepsilon## I mean to say that, no matter how small ϵ is, there is always an element of S whose distance from supS...

23. ### How to prove this statement about the derivative of a function

My try: ##\begin{align} \dfrac{d {r^2}}{d r} \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \tag1\\ \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \dfrac{1}{\dfrac{d r^2}{d r}}=\dfrac{p-a\cos\theta}{r} \tag2\\ \end{align}## By chain rule...
24. ### I Finite expansion of a fraction of functions

I am having a problem finding the right order above and below to find the finite expansion of a fraction of usual functions assembled in complicated ways. For instance, a question asked to find the limit as x approaches 0 for the following function I know that to solve it we must first find...
25. ### I Do I need induction to prove that this sequence is monotonic?

I think the initial assumptions would allow me to prove this without induction. Suppose ##(x_n)## is a real sequence that is bounded above. Define $$y_n = \sup\{x_j | j \geq n\}.$$ Let ##n \in \mathbb{N}##. Then for all ##j \in \mathbb{N}## such that ##j \geq n + 1 > n## $$x_{j} \leq y_n.$$...
26. ### I Rudin: theorem 1.21

Summary: Rudin theorem 1.21 He has said that as t=X/(X+1) then t^n<t<1 then maximum value of t is 1. then in the next part he has given that t^n<t<x. as maximum value of t is less than 1 why has he given that t<x ?
27. ### Help with a real analysis problem

I tried to prove this by absurd stating that there is no such ## \mu'## but i couldn't get anywhere...
28. ### A What type of function satisfy a type of growth condition?

Let ##f:\mathbb{R}^n\rightarrow\mathbb{R}^n##. Is there any class of function and some type of "growth conditions" such that bounds like below can be established: \begin{equation} ||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right), \end{equation} with ##\mathcal{X}:= \{x:f(x)=0\}## (zero...
29. ### Positive derivative implies growing function using Bolzano-Weierstrass

I'm stuck on a proof involving the Bolzano-Weierstrass theorem. Consider the following statement: $$f'(x)>0 \ \text{on} \ [a,b] \implies \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2$$ i.e. a positive derivative over an interval implies that the function is growing over the...
30. ### Function Continuity Proof in Real Analysis

Homework Statement We've been given a set of hints to solve the problem below and I'm stuck on one of them Let f:[a,b]->R , prove, using the hints below, that if f is continuous and if f(a) < 0 < f(b), then there exists a c ∈ (a,b) such that f(c) = 0 Hint let set S = {x∈[a,b]:f(x)≤0} let c =...

45. ### Regarding Real numbers as limits of Cauchy sequences

Homework Statement Let ##x\in\Bbb{R}## such that ##x\neq 0##. Then ##x=LIM_{n\rightarrow\infty}a_n## for some Cauchy sequence ##(a_n)_{n=1}^{\infty}## which is bounded away from zero. 2. Relevant definitions and propositions: 3. The attempt at a proof: Proof:(by construction) Let...
46. ### If A is dense in [0,1] and f(x) = 0, x in A, prove ∫fdx = 0.

Homework Statement "A set ##A\subset [0,1]## is dense in ##[0,1]## iff every open interval that intersects ##[0,1]## contains ##x\in A##. Suppose ##f:[0,1]\rightarrow ℝ## is integrable and ##f(x) = 0,x\in A## with ##A## dense in ##[0,1]##. Show that ##\int_{0}^{1}f(x)dx=0##." Homework...
47. ### B A Rational Game

This post is to set forth a little game that attempts to demonstrate something that I find to be intriguing about the real numbers. The game is one that takes place in a theoretical sense only. It starts by assuming we have two pieces of paper. On each is a line segment of length two: [0,2]...
48. ### Rudin POMA: chapter 4 problem 14

Homework Statement Question: Let ##I = [0,1]##. Suppose ##f## is a continuous mapping of ##I## into ##I##. Prove that ##f(x) = x## for at least one ##x∈I##. Homework Equations Define first(##[A,B]##) = ##A## and second(##[A,B]##) = ##B## where ##[A,B]## is an interval in ##R##. The Attempt at...
49. ### Convergence of a double summation using diagonals

Homework Statement Show that ##\sum_{k=2}^\infty d_k## converges to ##\lim_{n\to\infty} s_{nn}##. Homework Equations I've included some relevant information below: The Attempt at a Solution So far I've managed to show that ##\sum_{k=2}^\infty |d_k|## converges, but I don't know how to move...
50. ### A Solution of a weakly formulated pde involving p-Laplacian

Let $$f:\Omega\to\mathbb{R}$$, where $$\Omega\subset\mathbb{R}^d$$, and $$\Omega$$ is convex and bounded. Let $$\{x_i\}_{i=1,2,..N}$$ be a set of points in the interior of $$\Omega$$. $$d_i\in\mathbb{R},i = 1,2,..N$$ I want to solve this weakly formulated pde:  0=\frac{A}{N^{d+1}} \sum_i...