I assume that L is a propositional language and p1, p2, ... are all propositional variables. This information should be in the problem statement.

There is a general theorem that every consistent set of formulas has a maximal extension (i.e., superset). Let's call it Π. This fact is often used as a key lemma in proving model completeness of propositional (and first-order) logic. Since Π is maximal (i.e., no proper superset of it is consistent), it is complete. Indeed, suppose Π derives neither A nor ~A for some formula A. Then Π ∪ {A} is a proper superset of Π and it is consistent, for if Π and A derive a contradiction, then Π derives ~A contrary to our assumption.

You are left to show that {p2, p4, ...} ∪ {~p1, ~p3, ...} is consistent. But every derivation uses only a finite number of assumption, so if a contradiction is derived from this set, then a finite subset is also inconsistent. This would contradict the soundness theorem.

The theorem about maximal consistent extension is pretty simple. You just enumerate all formulas in the language and for each formula you add either it or its negation so that the set stays consistent. One has to prove that this is always possible. After that, the union of the increasing chain of sets is taken; it is easy to see that it is also consistent. In fact, this construction implies that the resulting set has either A or ~A for each formula A, i.e., it is complete.