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Cauchy Sequence Homework Hotline

Real analysis is an important area of mathematics that deals with sets and sequences of real numbers, as well as the functions of one or more real variables. It is one of the main branches of mathematical analysis. Real analysis can be treated as a subset of complex analysis, since many results of the former are special cases of results in the latter.

In real analysis, one studies the real numbers and real-valued functions of real variable, analytic properties of real functions and sequences, as well as convergence and limits of sequences of real numbers, calculus of the real numbers, continuity, and smoothness. It relates the properties of real-valued functions.

Real analysis is concerned with the behavior and properties of functions, sequences, and sets on the real number line, which is denoted mathematically by the letter R. The following concepts can be examined through real analysis: limits, continuity, derivatives, changing rates, and integration - the quantity of change which occurs over time.

On a conceptual or practical level, many of these ideas are covered at lower levels of mathematics, including calculus, therefore the subject of real analysis may seem rather senseless and trivial. Real analysis, however, deals with the depth, complexity, and beauty that is beneath the surface of everyday mathematics. There is an assurance of correctness which we call rigor that permeates the whole of mathematics. Thus, real analysis can, to some degree, be seen as the development of a rigorous, well-proven framework to support the ideas which are based on feelings rather than facts.

The major concepts in real analysis include the following:                                                        

  • Fundamental calculus notions including limits, continuity, derivatives, integrals, and the convergence and divergence of infinite series
  • Sequences of sets and unions and intersections of arbitrary numbers of sets
  • Least and greatest lower and upper bounds of a set
  • Elementary notions of topology, including open, closed, countable, connected, and compact sets
  • Liminf and limsup, respectively the "limit inferior" and "limit superior" of a sequence
  • Cauchy sequences and their relation to convergent sequences
  • Metrics and metric spaces, which generalize the notions of distance and Euclidean spaces
  • Pointwise convergence and uniform convergence of sequences of functions
  • Rates of convergence and "Big O notation"
  • Sigma algebras, measures and measure spaces

Real analysis is a very straightforward subject, as it is simply a nearly linear development of mathematical portion. Instead of relying on sometimes uncertain intuition (which we have all felt when we were solving a problem we did not understand), we will anchor it to a rigorous set of mathematical theorems.

Real analysis is one of the most important branches of mathematics.  If we want to understand differential equations or functional analysis or topology or complex analysis, we need to study the basic concept of real analysis.

 

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Do you want to help me with my homework? The exercise is as follows:

Consider a Lipschitz function, $h:\mathbb{R}\rightarrow\mathbb{R}$, satisfying for every $x, y$: $$\left| h(x)-h(y) \right| \leq \alpha \left| x-y \right|$$ with $0 \lt \alpha \lt 1$

  1. Show that there exists at most one $x$, with $h(x)=x$.
  2. Prove that $h$ is uniformly continuous.
  3. Take some $x_1\in\mathbb{R}$, and define inductively the sequence $(x_n)$ as $$ x_{n+1}=h(x_n), \quad n= 1, 2, \cdots$$ Show that for every $x_1$ the sequence $(x_n)$ is a Cauchy sequence.
  4. Take some $x_1$. Define $x= \lim x_n$. Show that $x=\lim{x_n}$ satisfies $h(x)=x$
  5. Show (using the first part), that the limits of the sequences $(x_n)$ for all choices of $x_1$ are all the same.

My work until now:

Part 1

Suppose $h(x_1)=x_1, h(x_2)=x_2$, and $x_1 \not = x_2$. Then, following the condition, $$\left| h(x_1)-h(x_2) \right|= \left| x_1-x_2 \right| \le \alpha \left|x_1-x_2\right|.$$ This means that $\frac{\left|x_1-x_2\right|}{\left|x_1-x_2\right|}=1 \le \alpha$. But $\alpha <1$ was given, so $x_1 = x_2$. There exists at most one $x$ with $h(x)=x$

Part 2

Let $\epsilon>0$ be given and choose $\delta=\frac{\epsilon}{\alpha}$. Then, for any $x, y$ with $\left|x-y\right|<\delta = \frac{\epsilon}{\alpha}$, I have $$\left| h(x)-h(y) \right| \le \alpha \left|x-y\right| \lt \alpha\left(\frac{\epsilon}{\alpha}\right)=\epsilon,$$ which shows that $f$ is uniformly continuous.

Part 3

I don't know where to start.

Part 4

real-analysissequences-and-seriescontinuityfixed-point-theorems

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