Could someone please explain the proof of Hotelling's lemma?
Well done for spotting this one. It is (was) simply a wrong statement. It is indeed the maximum and not the minimum of the expression as (was in the past) stated in the wikipedia article. 

Then the maximum of this expression is the maximum of $-C$ (Cost) which is equivalent to the minimum of $C$, and the equivalence with maximum profit holds under duality.