Contraction mapping theorem
Let be a function defined on all of , and assume there is a constant such that and for all .
a) Show is continuous on .
Let be a real number and . Then by definition. So, let . It follows that implies .
b) Pick some point and construct the sequence
In general, if , show that the resulting sequence is a Cauchy sequence. Hence, we may let .
I am not sure how to approach this one. Some help getting started would be appreciated. It seems almost trivial, but I am not sure how formalize it.