I am looking at the proof of Bishop's Theorem on pages 122 and 123 of Rudin's Functional Analysis. The following quote is from the the last two sentences of the proof on pg. 123.
"Every continuous linear functional on $C(S)$ that annihilates $A$ also annihilates g. Hence $g\in A$ by the Hahn-Banach Separation Theorem."
I understand the first sentence but I don't see why that and the HBST implies $g\in A$.