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Real Analysis

Real Analysis Proof of a Function

If f is a function from R to R, and there exists a real number aE(0,1) such that |f'(x)|≤a for all xER , show that the equation x = f(x) has a solution.

Real Analysis: Show an Integral Equation Has a Unique Solution

Assume that g(t) is continuous on [a,b], K(t,s) is continuous on the rectangle a≤t, s≤b and there exists a constant M such that (a≤s≤b). Then the integral equation has a unique solution when . Please see the attached file for the fully formatted problems.

Real Analysis: Compact Space and Infimum

Assume that f is a continuous real valued function on the compact space X, then show there exists a point x-bar E X such that f(x-bar)=inf{f(x): x E X). See the attached file.

Testing Series for Convergence

Test for convergence or divergence 1.) sum from n=1 to infinity of (e^1/n)/(n^2) 2.) sum from j=1 to infinity of (-1)^j * ((sqrt j)/(j+5)) 3.) sum from n=2 to infinity of (1/((1+n)^(ln n)) keywords: tests

Show function defines a metric space and the space is complete

Let X be the set of all continuous functions from I_1=[t_0-a_1, t_0+a_1] into the closed ball B[g(t_0);b] is a subset of R_n. Show that for each a>0 the rule d(x,y)=max(|x(t)-y(t)|e^(-a|t-t_0|)) defines a metric on X and that the metric space (X,d) is complete.

Real Analysis - Banach Fixed Point Theorem

Prove the following generalization of the Banach Fixed Point Theorem: If T is a transformation of a complete metric space X into itself such that the nth iterate, T^n, is a contraction for some positive integer n, then T has a unique fixed-point.

Real Analysis - Newton's Method and showing convergence.

Newton's Method: Consider the equation f(x)=0 where f is a real-valued function of a real variable. Let x_0 be any initial approximation of the solution and let x_(n+1)=x_n - (f(x_n)/f'(x_n)). Show that if there is a positive number "a" such that for all x in [x_0-a, x_0+a] |(f(x)f''(x))/((f'(x))^2)|<=lambda<1 and |(f(x

Real Analysis - Show E is Equicontinuous

Let E be a set of differentiable functions in C[a,b] with uniformly bounded derivatives; i.e., there exists a number M, independent of f in E, such that |f'(x)|<=M for all x in [a,b] and all f in E. Show that E is equicontinuous.

Continuous function on compact space

Show that if f is a continuous real-valued function on the compact space X, then there exist points x_1, x_2 in X such that f(x_1)=inf{f(x):x in X} and f(x_2)=sup{f(x):x in X}.

Real Analysis : Bounded Open Balls

Show that a set E in the metric space X is bounded if and only if, for some "a" in X, there exists an open ball B(a;r) such that E is a subset of B(a;r).

Real analysis - open and closed sets

(See attached file for full problem description) 1. In the metric space show that: a. Any open interval of the form (a,b), (a, ), or (- ,b) is an open set. b. A close interval [a,b] is a closed set. c. Any interval of the form [a, ) is a closed set.

Real analysis metric spaces

(See attached file for full problem description) 7. If d is a real-valued function on which for all x, y, and z in X satistifes d(x,y) = 0 if and only if x=y d(x,y)+d(x,z)&#8805;d(y,z) show that d is a metric on X.

Real analysis metric spaces

(See attached file for full problem description) Let A be any set and let X be the set of all bounded real-valued functions defined on A. Show that defines a metric on X.

Real Analysis Metric Spaces

(See attached file for full problem description) Let X be the set of all bounded sequences of real numbers. If and are elements of X, show that the function d defined by is a metric on X.

limit and solving trigonometric equation

(See attached files for full problem description) 1. Find the limit: lim(t-->0) t^2/(1-cost) 2. solve the following trigometric equation tan(2x) = 2sin(x), where 0<=0< 360 degrees