Unfortunately I have to say that practically nothing of what you say is correct. In particular, your negation of a sequence an being a Cauchy sequence is wrong. A sequence is not Cauchy if and only if
The problem is "standard"; it is fundamental in proving a "fixed point" theorem. Here is a reference to this fixed point theorem which contains essentially a proof of your problem -- Banach fixed-point theorem - Wikipedia, the free encyclopedia. You want to look at Banach's original proof.