Proposition 6

Let $\mathfrak{F}$ be a filter on a set $X$, and let $f$ be a mapping of $X$ into a complete uniform space $X'$. Then $f$ has a limit with respect to $\mathfrak{F}$ if and only if the image of $\mathfrak{F}$ under $f$ is a Cauchy filter base.

This criterion shows the importance of complete spaces in all questions involving the notion of limit: if a function takes its values in a complete space we can prove the existence of a limit without knowing in advance the value of the limit; this would be impossible if the definition of limit were the only criterion of convergence at our disposal.

A uniformity which is finer than the uniformity of a complete space need not be a uniformity of a complete space (Exercise 2). However, we have the following proposition: