AHSME 1967 Problem 20

A circle is inscribed in a square of side $m$, then a square is inscribed in that circle, then a circle is inscribed in the latter square, and so on. If $S_n$ is the sum of the areas of the first $n$ circles so inscribed, then, as $n$ grows beyond all bounds, $S_n$ approaches: $\textbf{(A)}\ \frac{\pi m^2}{2} \textbf{(B)}\ \frac{3\pi m^2}{8} \textbf{(C)}\ \frac{\pi m^2}{3} \textbf{(D)}\ \frac{\pi m^2}{4} \textbf{(E)}\ \frac{\pi m^2}{8}$