\begin{definition}[Definition:Limit Inferior of Moore-Smith Sequence]
Let $\left({S, \preceq}\right)$ be a directed set.
Let $L = \left({T, \precsim}\right)$ be a complete lattice.
Let $N: S \to T$ be a Moore-Smith sequence in $T$.
Then '''limit inferior''' of $N$ is defined as follows:
:$\liminf N := \sup_L \left\{ {\inf_L \left({N \left[{\preceq \left({j}\right)}\right]}\right): j \in S}\right\}$
where
:$\preceq \left({j}\right)$ denotes the image of $j$ by $\preceq$,
:$N \left[{\preceq \left({j}\right)}\right]$ denotes the image of $\preceq \left({j}\right)$ under $N$.
\end{definition}