Finding least upper bound in meet-semilattice

Question

Let $(L,\leq)$ be a meet-semilattice with greatest element $1 \in L$. Suppose that every bounded descending and ascending chain in $L$ stabilizes. Is it possible to prove that for every pair of elements $a,b \in L$, there exists a least upper bound, that is, $(L,\leq)$ is a lattice? Every pair $a,b \in L$ has obviously $1$ as upper bound. Shouldn't $L$ be totally ordered to apply the chain conditions to find a least upper bound?

Answer 1

Hint: Let $A$ be the set of common upper bounds to $a$ and $b$.
As you noted, $1 \in A$, so that $A \neq \varnothing$.
Now use use that descending chain to prove that there is $u \in L$ such that $\bigwedge A = u$.
Then prove that $u = \sup\{a,b\}$.