Cauchy criterion for the existence of a limit of a function

Theorem 1.

Let $S$ be a set and $B$ a filter basis in $S$. A function $f\mathrm{:}S\mathrm{\to }\mathrm{R}$ possesses limit on $B$, iff for every $ϵ\mathrm{>}\mathrm{0}$ there exists $X\mathrm{\in }B$ such that the oscillation of $f$ on $X$ is less than $ϵ$.

For a proof of this theorem see[1].