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 lipschitz condition

(See attached file for full problem description)

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].