Suppose $(P,\leq)$ is a generic lattice. Then for every $\{a,b\}$ in the relation $ \leq $, the set of its upper bounds is not empty and has a least element, and the set of its lower bounds is not empty and has a least element.
Suppose $ S $ is a non-empty subset of the lattice. Two cases arise:
we can denote the upper bounds and lower bounds of a set with $u(S)$ and $ l(S)$ respectively
if $\{a,b\}$ is included in $ S $, then $u(S) $ is included in $ u(\{a,b \})$ since if $x$ is in $u(S) $, then it is true that $ a \leq x$ and $ b \leq x$. But since $ u(\{a,b \})$ has least element, every set included in it also has a least element, therefore there is a least upper bound also in $ u(S)$
The reasoning for the lower bound is analoguous.
I do not understand how to prove it for { a } included in { a,b }. Could you possibly give me some hint on how to proceed with the solution?
Thanks in advance