TITLE: Completeness: Nets vs. Sequences
QUESTION [1 upvotes]: Prove that a metric space is complete w.r.t. sequences iff it is complete w.r.t. nets!
(The converse is trivial of course!)

REPLY [2 votes]: Consider a Cauchy net:
$$\forall \lambda,\lambda'\geq\lambda_n:\quad d(x_\lambda,x_\lambda')<\frac{1}{n}$$
Extract a Cauchy sequence:
$$x_n:=x_{\lambda(n)}\quad\lambda(n):=\lambda_1\wedge\ldots\wedge\lambda_n$$
Apply completeness:
$$d(x_\lambda,x)\leq d(x_\lambda,x_{n_0})+d(x_{n_0},x)<\frac{N}{2}+\frac{N}{2}\leq\epsilon$$
where to choose the meet $n_0:=N\wedge n(N)$ with $N:=\lceil\frac{\epsilon}{2}\rceil$
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