Suppose that $a_1, a_2, \cdots$ are positive real numbers. Prove that
$$\sum_{n=1}^{\infty} (a_1 a_2 \cdots a_n)^{1/n} \leq e \sum_{n=1}^{\infty} a_n \,.$$

The best solution was submitted by 정성진. Congratulations!

Alternative solutions were submitted by 김경석 (+3), 이영민 (+3), 이종원 (+3), 장기정 (+3), 정성진 (+3), 조준영 (+3), 황성호 (+2). Incorrect solutions were submitted by K.S.J., L.S.C. (Some initials here might have been improperly chosen.)