Longest increasing subsequence

[Naumberg]

Problem:

"Let $x_1, ..., x_n$ be i.i.d random variables uniformly on [0,1]. Let $X$ be the length of the longest increasing subsequence of $x_1, ..., x_n$. Show that $E[X] \ge (1-o(1))(1-e^{-1}) \sqrt{n}$."

Hi forum!

Using the Erdos' lemma I can only deduce that $E[X] \ge \frac{1}{2} \sqrt{n}$, which is a weaker bound unfortunately.

I would appreciate any further ideas!

Thanks for your help,
Michael

 

[Stephen Tashi]

What does the notation "$o(1)$" mean in this context?