Set of Subgroups forms Complete Lattice/Proof 1

Proof
Let $\O \subset \mathbb H \subseteq \mathbb G$.

By Intersection of Subgroups: General Result, $\bigcap \mathbb H$ is the largest subgroup of $G$ contained in each of the elements of $\mathbb H$.

Thus, not only is $\ds \bigcap \mathbb H$ a lower bound of $\mathbb H$, but also the largest, and therefore an infimum.

The supremum of $\mathbb H$ is the smallest subgroup of $G$ containing $\bigcup \mathbb H$.

Therefore $\struct {\mathbb G, \subseteq}$ is a complete lattice.