Let $\{a_n\}$ denote sequence in metric space X. We have well known classical definition of Cauchy sequence:
$ \forall_{\epsilon>0}\exists_N\forall_{n,m>N}d(a_n,a_m)<\epsilon $
We have also definition in terms of entourages in uniform space(adapted to a sequence):
$ \forall_{U\in\Phi}\exists_N\forall_{n,m>N}(a_n,a_m)\in U $
where canonical uniformity $\Phi$ of metric space is defined here.
It seems quite reasonable to me that these two definition should coincide on given setting, but I couldn't find a proof and so far also couldn't prove it myself.
So, what is the proof? And also is it of any significance that we are dealing with sequences instead of nets? That is we would replace the sequence $\{a_n\}$ with a net $\{a_\alpha\}$, would the equivalence between classical and uniform definitions still follow?