Infinite set, disjunction, and tautology

[iambored]

Let {A1, A2, A3, ... } be an infinite set of formulas in propositional logic. Assume that
for every valuation v there is some n (depending on v) such that v(An) = 1. Show
then that there is some fixed m with A1 \vee A2 \vee ... \vee Am a tautology.

This is equivalent to showing that v(Ai) = 1 for at least one 1\leqi\leqm. But I'm not sure where to proceed from here.

[JSuarez]

Start by considering the set \left\{A_1,A_1\vee A_2,A_1\vee A_2 \vee A_3\cdots\right\} and assume that none of its elements is a tautology. Note also that any valuation may be identified with an infinite binary sequence.