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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

Sequences and Series (20 Problems): Partial Sums, Convergence and Divergence

Please do all problems below step by step showing me everything. Do simply as possible so I can clearly understand without rework. Adult here relearning so show all work, etc. OK, some said cannot read problems, but do not have a scanner with me know, so typed them in below. Sorry for any problems, but this shopuld clear up

Finding Limits and Applying the L'Hopital's Rule

Find the indicated limit make sure you have a indeterminate form before you apply L'Hopital rule (1) lim xgo to0 arctan3x/arcsinx (2) lim x go to pi/2 3secx+5/tanx (3)lim x go to 0 2csc^2x/cot^2x evaluate dx/squrtpix a=0 andb=inifinity.