Equivalence of Definitions of Complete Metric Space

Theorem
The two definitions of complete metric space:


 * $(1):\quad$ A metric space $\left({X, d}\right)$ is complete if every Cauchy sequence is convergent


 * $(2):\quad$ A metric space $\left({X, d}\right)$ is complete iff the intersection of every nested sequence of closed balls whose radii tend to zero is non-empty

are logically equivalent.