Prove the $C(A) = l(u(a))$ is a closure operator on $A$

Question

Let $L$ be a lattice. For $A \subseteq L$ define

$u(A) = \{b \in L : a \le b, a \in A \}$ and $l(A) = \{b \in L : b \le a, a \in A \}.$

I want to show that $C(A) = l(u(A))$ is a closure operator on $L$.

I understand the problem, but I am not sure how to write it up. For it to be a closure operator I need to show that

i) $X \subseteq C(X)$.

ii) $X \subseteq Y \implies C(X) \subseteq C(Y)$.

iii) $C(C(X)) = C(X)$.

I am not sure if I need to do it by some sort of induction or just on elements of $L$.

Answer 1

I think you are overthinking this. The basic requirements of reflexivity and transitivity of $\leq$ are all you need to prove it.