Increasing sequence of positive operators

Question

Denote the set $\mathcal C:=\{x\in\mathbf A:0\leq x\leq p\}$ where $\mathbf A$ is $C^*$ algebra and $p$ is a projection. How to show that every chain in $\mathcal C$ has a maximal element? Actually, I need to use Zorn's lemma in some other context but this where the case reduces to.