Limit Inferior of Repetition Net

Theorem
Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice.

Let $N = \left({\N, \le}\right)$ be a directed ordered set.

Let $a, b \in S$.

Let $f = \left({c_i}\right)_{i \in \N} = \left({a, b, a, b, \dots}\right):\N \to S$ be a Moore-Smith sequence.

Then $\liminf \left({c_i}\right)_{i \in \N} = a \wedge b$

Proof
We will prove that
 * (lemma): $\forall j \in \N: f\left[{\le\left({j}\right)}\right] = \left\{ {a, b}\right\}$

Let $j \in \N$.

Let $x \in S$.

Assume that
 * $x \in f\left[{\le\left({j}\right)}\right]$

By definition of image of set:
 * $\exists i \in \le\left({j}\right): x = f\left({i}\right)$

By definition of $f$:
 * $x = a$ or $x = b$

Thus by definition of unordered tuple:
 * $x \in \left\{ {a, b}\right\}$

Assume that
 * $x \in \left\{ {a, b}\right\}$

We have:
 * $j \le j$ and $j \le j+1$

By definition of image of element:
 * $j, j+1 \in \le\left({j}\right)$

By definition of $f$:
 * $\left\{ {f\left({j}\right), f\left({j+1}\right)}\right\} = \left\{ {a, b}\right\}$

By definition of image of set:
 * $f\left({j}\right), f\left({j+1}\right) \in f\left[{\le\left({j}\right)}\right]$

Thus by definition of unordered tuple: $x \in f\left[{\le\left({j}\right)}\right]$

Thus by definition of set equality:
 * $f\left[{\le\left({j}\right)}\right] = \left\{ {a, b}\right\}$

Thus