Let $X_1,X_2,X_3,...$ be well formed formulas. If for every truth assignment $v$ there exists $n$ with $X_n$ satisfied by $v$, show there exists $n$ with $X_1\lor...\lor X_n$ a tautology.
We can assume there are an infinite number of sentence symbols, as a finite number would imply a finite number of truth assignments, so we could take $n$ to be the maximum of the first satisfying indices of the truth assignments. I think it's important that the number of sentence symbols is countable. For all $n$, let $Y_n=X_1\lor...\lor X_n$ and $S_n$ be the set of truth assignments satisfying $Y_n$. Note that $S_1\subseteq S_2\subseteq S_3\subseteq...$. Since the number of sentence symbols is countable and every truth assignment sends every sentence symbol to $0$ or $1$, the truth assignments are in bijection with countable binary sequences, which are in bijection with the real numbers between $0$ and $1$. I was hoping to get a contradiction out of this, but we could just make $S_n\setminus S_{n-1}$ the binary numbers from $0$ to $1$ with $n$ leading $0$s. Compactness of well formed formulas might be relevant, but I can't think of a way to apply it.