Equivalence of Definitions of Complete Metric Space

Theorem
The two definitions of complete metric space are logically equivalent:

Proof
Definition 1 implies definition 2 by the Nested Sphere Theorem.

Suppose that $\left({X, d}\right)$ is a complete metric space by definition 2.

Let $\left\langle{x_n}\right\rangle$ 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 $n, m \in \N$:
 * $n, m \ge N_k \implies d \left({x_n, x_m}\right) < \dfrac 1 {2^k}$

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