  Given a sequence of $n$ independent random variables with common continuous
distribution, we propose a simple adaptive online policy that selects a
monotone increasing subsequence. We show that the expected number of monotone
increasing selections made by such a policy is within $O(\log n)$ of optimal.
Our construction provides a direct and natural way for proving the $O(\log
n)$-optimality gap. An earlier proof of the same result made crucial use of a
key inequality of Bruss and Delbaen (2001) and of de-Poissonization.
