Completeness of R^2 with sup norm
Bugatti79, you need to rethink your entire approach to proofs. Theorems are always implications, i.e. statements of the form ##A\Rightarrow B##, but you always ignore that. You always try to avoid making the assumption A, you always try to avoid using the definitions of the terms in A, and most of the time, you even assume B! These are the three biggest mistakes that can possibly be made in a proof. You also often make irrelevant assumptions that have nothing to do with the theorem.
You want to prove that if a sequence is Cauchy with respect to the ∞-norm, it's convergent with respect to the ∞-norm. So A is the statement "##\langle x_n\rangle## is Cauchy with respect to the ∞-norm", and B is the statement "##\langle x_n\rangle## is convergent with respect to the ∞-norm". And you start by assuming B, as usual. This is the single biggest mistake that can be made in a proof.
One thing you need to understand is that once a proof of ##A\Rightarrow B## has arrived at the statement B, there's nothing more to say. That's the end of the proof. So if you start by assuming B, nothing more needs to be said. In fact, it wouldn't make any sense to say anything more after that.