Real Analysis - contraction is continuous

Show that a contraction is continuous.

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 ...continues

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: 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 : Fredholm equation and Lipschitz condition

Consider the nonlinear Fredholm equation where is continuous on [a,b] and is continuous and satisfies a Lipschitz condition: on the set . Show that the integral equation has a unique solution on [a,b] if .

Show that a rule is a metric

Show that a rule is a metric. See attached file for full problem description.

Real Analysis : Riemann Integrals

If f is a function from R to R which is increasing on [a,b], show that f is Riemann integrable on [a,b].

Real Analysis : Riemann Integrals

Let f be a function from R^n to R which is bounded on [a,b]. Show that f is Riemann integrable on [a,b] if and only if for each epsilon>0 there is a partition P such that U(f,P)-L(f,P)

Proof using Riemann Integrals

If and are functions from to which are Riemann integrable on and which differ at only a finite number of points in , show that . Please see the attached file for the fully formatted problems.

Proof : Show integral is zero.

If Xc is the function from to such that Xc(x)={1 if x=c and cE[a,b] , show that ∫a-->b Xc =0. {0 if x≠c Please see the attached file for the fully formatted problems.