Completeness: Nets vs. Sequences

Question

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!)

Answer 1

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 $N:=\lceil\frac{\epsilon}{2}\rceil$