Hi! I might have found something wrong with 1.2. I hope my reasoning below isn't all wrong, but here goes:
A part of 1.2 is to prove that if a sequence is Cauchy in (X, d_1), then it is convergent in (X, d_2), where d_1 and d_2 are equivalent metrics. But d_1 is equivalent to itself. So a corollary would be that if sequence is Cauchy in X, then it is convergent in X. Which is not the always the case?