Infinitary Logic Has No Expressive Efficiency Over Finitary Logic

We can measure the complexity of a logical formula by counting the number of alternations between existential and universal quantifiers. Suppose that an elementary first-order formula $\phi$ (in ${\mathcal{L}}_{\omega ,\omega }$) is equivalent to a formula of the infinitary language ${\mathcal{L}}_{\mathrm{\infty },\omega }$ with $n$ alternations of quantifiers. We prove that $\phi$ is equivalent to a finitary formula with $n$ alternations of quantifiers. Thus using infinitary logic does not allow us to express a finitary formula in a simpler way.