Is the opposite of the Cantor's Intersection Theorem True?

Question

I was trying to prove that in a metric space, if any sequence of nested closed metric balls such that their radius converges to zero has non-empty intersection, then the space is complete. But I couldn't.

Answer 1

If a metric space $X$ is not complete then there exists a Cauchy sequence $(a_n)$ that does not converge. Let $\bar{X}$ be the completion of $X$ and let $a \in \bar{X} \setminus X$ be the limit of the sequence $(a_n)$. Pass to a subsequence $(a_m)$ such that $d(a_m, a)$ is decreasing. Then the sequence of nested closed sets $$ A_m = \{x \in X : d(x, a) \leq d(a_m, a)\} \subset X$$ has empty intersection and their radius converges to zero.

Edit: As pointed out in the comments, this does not really answer your question since you were asking for balls, and not just closed sets. To obtain balls from this construction, I guess one should choose the same subsequence, but instead consider balls with very carefully chosen radii around the points in the sequence. The sets $$ B_m = \{x \in X : d(x, a_m) \leq d(a_m , a) + 2^{-n}\}$$ should work, if I am not mistaken again. Sorry for the confusion.