A metric space $X$ is said to be complete if every Cauchy sequence in $X$ has a limit which is an element of $X$.
If a metric space lacks Cauchy sequences, e.g. $\mathbb{N}\subset\mathbb{R}$, is it then complete?
A metric space $X$ is said to be complete if every Cauchy sequence in $X$ has a limit which is an element of $X$.
If a metric space lacks Cauchy sequences, e.g. $\mathbb{N}\subset\mathbb{R}$, is it then complete?
Well, yes, but in fact the only metric space with no Cauchy sequences is the empty metric space. $\mathbb{N}$ has plenty of Cauchy sequences, for example $1,1,1,1,1,\ldots$ and $1,2,3,4,5,57,57,57,57,57,\ldots$.