Problem1 Example& Counterexample

a) Give an example of a set which is not closed but is a countable union of closed sets

b) Provide a counterexample for the following statement: For two uniformly continuous functions f and g onR, where f or g is not bounded, f *g is uniformly continuous.

Problem 2 The following statement is called the Contraction Mapping Theorem.

Let f be a function on R, and assume that there sxists a k belong to | f(x)-f(y)|<=k|x-y|,for all x,y belong to R: such an f is called a contraction mapping. Then f is continuous, and there exists a unique a belong to R such that f(a)=a i.e.a (unique) fixed point.

What follows now is the proof of the contraction mapping theorem. Each step has a conclusion, in bold, provide the argument for the conclusion.Step1-step6

Sample Solution