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 \(\displaystyle \lim_{n\to \infty} e_n=\frac1e\).