Up-Complete Lower Bounded Join Semilattice is Complete

Theorem
Let $\left({S, \preceq}\right)$ be an up-complete lower bounded join semillattice.

Then $\left({S, \preceq}\right)$ is complete.

Proof
Let $X$ be a subset of $S$.

Define
 * $Y := \left|[ {\sup A: A \in {\it Fin}\left({X}\right) \land A \ne \varnothing}\right\}$

where ${\it Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.

By Existence of Non-Empty Finite Suprema in Join Semilattice
 * all suprema in $Y$ exist,

We will prove that
 * $Y$ is directed