Putnam 1965 A3

A-3. Show that, for any sequence $a_{1}, a_{2}, \cdots$ of real numbers, the two conditions (A) $$\lim _{n \rightarrow \infty} \frac{e^{\left(i a_{1}\right)}+e^{\left(i a_{2}\right)}+\cdots+e^{\left(i a_{n}\right)}}{n}=\alpha$$ and (B) $$\lim _{n \rightarrow \infty} \frac{e^{\left(i a_{1}\right)}+e^{\left(i a_{4}\right)}+\cdots+e^{\left(i a_{n}^{2}\right)}}{n^{2}}=\alpha$$ are equivalent.