Closure operator in complete lattices

Question

I want to prove that given a complete lattice $L $ the function $C : \mathcal{P}(L) \rightarrow \mathcal{P}(L)$ that assigns to $X \in L$ the set $\{a\in L : a\leq \bigvee X\ = Sup\{X\} \} $ is idempotent i.e. $C^2(X) =C(X)$.

I tried it proving that $sup\{X\} = sup\{C(X)\}$ but only could prove that $sup\{X\} \leq sup\{C(X)\}$

Any suggestion will be useful, thanks.

Answer 1

You proved that $\bigvee C(X)$ is an upper bound of $X$.
Suppose now that $u$ is another upper bound of $X$ and show that $\bigvee C(X) \leq u$.

To do that, pick $a \in C(X)$. By definition, $a \leq \bigvee X \leq u$, and so $a \leq u$.