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.

Proof
$(1)$ implies $(2)$ by the Nested Sphere Theorem.

Suppose that $(X,d)$ is a complete metric space by definition $(2)$.

Let $\langle{x_n}\rangle$ be a Cauchy sequence in $X$.

By the definition of a Cauchy sequence, for each $k \in \N_{>0}$, there is an $N_k \in \N_{>0}$ such that for all $n,m \in \N$:
 * $n,m \ge N_k \implies d(x_n,x_m) < \dfrac 1 {2^k}$

For each $k \in \N_{>0}$, let $C_k = B^-_{1/2^k} \left({x_{N_k}}\right)$.