I fear I over simplified the following problem:
For any partially ordered set $(A,\leq)$, let $A^* = A- \{\max A,\min A\}$ if $\max A$ and $\min A$ exist. Show the inclusion $(A^*,\leq)\hookrightarrow (A,\leq)$ is $\sup$-continuous.
So I took any set $B\subseteq A^*$ such that $\sup B$ exists. Then $\iota(B)=B$, and $\iota(\sup B)=\sup B$. Then $\iota(\sup B)=\sup B=\sup(\iota(B))$, and so $\iota$ is $\sup$-continuous.
This seems too simple so I'm sure I've misinterpreted something. Can someone point out the source of error?