Alternative approach to prove that if a sequence is a Cauchy Sequence, then it converges.

For (i), simply let $b_n:=a_{n+1} ,\forall n\ge1.$ Then by the assumption, $(b_n)$ is a convergent subsequence of $(a_n)$ , and it implies obviously that $(a_n)$ is a convergent sequence by the definition of limits.

For (ii), try to argue by contradiction with the definitions of convergent sequences, limits, and Cauchy sequences.

The basic idea is that, if a sequence $(a_n)$ is divergent, then no matter how rear in the "tail"(i.e. $\{a_n\,\vert n\ge k, k$ large$\}$, there always exist two elements of it which are not close enough. However, every two terms with sufficiently large indexes of a Cauchy sequence can get sufficiently close. This is the contradiction.