My attempt: Assume that in $(X, d)$ the Cantor intersection theorem holds for all closed sets.
Assume for the sake of contradiction that $(X, d) $ is not complete then we have a Cauchy sequence $\{x_n\}$ is a Cauchy sequence which is not convergent.
Then the set $F_1= \{x_n\}$ is a closed set as $F_1$ doesn't have a limit point (if it does have a limit point then the sequence will be convergent).
Let $F_2= \{x_n\} - x_1$ and induction we can have $F_n = \{x_n\} - x_{n-1}$ so that each $F_n$ is closed and it is a nested sequence of closed sets.
Also by using the property of Cauchy sequence we can see that $diam(F_n) \to 0$. Hence $\cap_{n \ge 1}F_n$ isnonempty and contains a singleton element $x$ which is the limit of the sequence.
Hence the Cauchy sequence is convergent.
The other part is given in the book. Can someone go through this proof?