Proposition 1

Let $f$ and $g$ be two real-valued functions defined on a set $X$ filtered by a filter $\mathfrak{F}$. If $\lim_{\mathfrak{F}} f$ and $\lim_{\mathfrak{F}} g$ exist, and if, for each subset $A \in \mathfrak{F}$, there exists $x \in A$ such that $f(x) \leq g(x)$, then we have $\lim_{\mathfrak{F}} f \leq \lim_{\mathfrak{F}} g$.

To prove this result we shall prove the following equivalent statement: