Limit Inferior of Repetition Moore-Smith Sequence

Theorem
Let $L = \struct {S, \wedge, \preceq}$ be a meet semilattice.

Let $N = \struct {\N, \le}$ be a directed ordered set.

Let $a, b \in S$.

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

Then $\liminf \sequence {c_i}_{i \mathop \in \N} = a \wedge b$

Proof
We will prove that
 * (lemma): $\forall j \in \N: f \sqbrk {\le \paren j} = \set {a, b}$

Let $j \in \N$.

Let $x \in S$.

Assume:
 * $x \in f \sqbrk {\le \paren j}$

By definition of image of set:
 * $\exists i \in \le \paren j: x = \map f i$

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

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

Assume:
 * $x \in \set {a, b}$

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

By definition of image of element:
 * $j, j + 1 \in \le \paren j$

By definition of $f$:
 * $\set {\map f j, \map f {j + 1} } = \set {a, b}$

By definition of image of set:
 * $\map f j, \map f {j + 1} \in f \sqbrk {\le \paren j}$

Thus by definition of unordered tuple: $x \in f \sqbrk {\le \paren j}$

Thus by definition of set equality:
 * $f \sqbrk {\le \paren j} = \set {a, b}$

Thus: