Proving questions about posets

Question

How do I tackle a proof about posets? I have know idea how to approach this problem. Thanks!

Prove that if all subsets of a poset P have least upper bounds, then all subsets of P have greatest lower bounds.

Answer 1

Hint: Let $(P, \le)$ denote your poset and $A \subseteq P$. We want to have a greatest lower bound for $A$, so we consider the set $B = \{p \in P \mid p \le A\}$ of all lower bounds. By assumption, there is a lowest upper bound $p$ for $B$. How does it relate to $A$?