Let e_n be the expect value of the product x₁x₂ …x_n where x₁ is chosen uniformly at random in (0,1) and x_k is chosen uniformly at random in (x_k-1,1) for k=2,3,…,n. Prove that \(\displaystyle \lim_{n\to \infty} e_n=\frac1e\).

The best solution was submitted by Seong-Gu Jeong (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2009-20.