Real Analysis : Integral Proof

If X(c,d) is the function from into such that X(c,d)(x)={1 if xE(c,d) and (c,d)⊂[a,b], show that ∫a-->b Xc,d =d-c. {0 otherwise Please see the attached file for the fully formatted problem.

Cauchy sequence is a bounded sequence

Prove that a Cauchy sequence is a bounded sequence.

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) .

Real Analysis: Infinite union of compact sets.

Is the infinite union of compact sets compact? Is so please prove why if not please explain.

Real Analysis : Equicontinuous

If E is equicontinuous in C(X,R), show that E-bar (the closure of E) is also equicontinuous. keywords: equicontinuity

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 Proof : 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.

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.

Show that the given type of function on a compact metric space has a unique fixed point.

Assume that (X, d) is a compact metric space, and let f: X -> X be a function such that the inequality d(f(x), f(y)) < d(x, y) holds for all distinct elements x, y in X. Show that f has a unique fixed point. See attached file for full problem description.

Real Analysis: Use lower/upper integral to determine Riemann integrability

Let be defined by . Use lower integral and upper integral to determine the Riemann integrability of f on [0,1]. Please see the attached file for the fully formatted problems.

Real Analysis: Lipschitz Continuous

Prove that a Lipschitz continuous function f: X--> X is always continuous. How about the converse case? keywords: continuity