I am currently trying to proof that the power set of A is a complete lattice.
Since $\mathcal{P}(A),\subset$ is a partially ordered set, we still have to proof that $\sup(X)$ and $\inf(X)$ exist, for every not empty subset of $\mathcal{P}(A)$.
One can see, making a sketch that:
$$
\sup(X)=\cup_{C \in X} C
$$
$$
\inf(X)=\cap_{C \in X} C
$$
It is easy to proof that $\cup_{C \in X} C$ is a lower bound and that $\cap_{C \in X} C$ is an upper bound.
I don't get how to proof that they are the largest lower bound and the smallest upper bound.
Best Answer
Let $\mathcal X\subseteq\wp(A)$ and let $B$ be an upperbound of $\mathcal X$ w.r.t. the inclusion.
That means that $C\subseteq B$ for each $C\in\mathcal X$.
Then also $\bigcup\{C\mid C\in\mathcal X\}\subseteq B$ (do you see why?).
Next to that $\bigcup\{C\mid C\in\mathcal X\}$ itself is also an upper bound of $\mathcal X$.
(In your question you call it incorrectly a lower bound)
(To avoid confusion: $\bigcup\{C\mid X\in\mathcal X\}=\bigcup_{C\in\mathcal X}X$)
These two facts allow the conclusion that $\bigcup\{C\mid C\in\mathcal X\}$ is the least upper bound of $\mathcal X$.
Likewise it can be shown that $\bigcap\{C\mid C\in\mathcal X\}$ serves as greatest lower bound of $\mathcal X$.