Let $\displaystyle X_{n+1}\subset X_n$ for each $\displaystyle n\in\mathbb N.$ Let $\displaystyle (M,d)$ be a metric space. Prove that the following assertions are equivalent:

a)$\displaystyle (M,d)$ is a complete metric space.

b) For each $\displaystyle X_{n+1}\subset X_n$ being closed and non-empty where $\displaystyle \text{diam}(X_n)\to0$ as $\displaystyle n\to\infty,$ we have $\displaystyle \bigcap_{n=1}^\infty X_n\ne\varnothing.$

I already did a) $\displaystyle \implies$ b) but for the other implication, I have a long solution, is there a short way to do it?