# Lattice and Sublattice

I understand the definition of Lattice and Sublattice, but I cannot show the following exercise, since it seems trivial to me.

Show that, for any subset $$S$$ of a lattice $$L$$, the set $$s^{l}$$ of all the lower bounds of $$S$$ is a sublattice of $$L$$.

It's indeed easy: if $$a$$ and $$b$$ are lower bounds for all $$s\in S$$, that means $$a\le s, \, b\le s$$.
Then clearly also $$a\land b\le s$$ as e.g. $$a\land b\le a$$, and $$a\lor b\le s$$ by the definition of least upper bound ($$\lor$$).