  Consider a sequence of $n$ independent random variables with a common
continuous distribution $F$, and consider the task of choosing an increasing
subsequence where the observations are revealed sequentially and where an
observation must be accepted or rejected when it is first revealed. There is a
unique selection policy $\pi_n^*$ that is optimal in the sense that it
maximizes the expected value of $L_n(\pi_n^*)$, the number of selected
observations. We investigate the distribution of $L_n(\pi_n^*)$; in particular,
we obtain a central limit theorem for $L_n(\pi_n^*)$ and a detailed
understanding of its mean and variance for large $n$. Our results and methods
are complementary to the work of Bruss and Delbaen (2004) where an analogous
central limit theorem is found for monotone increasing selections from a finite
sequence with cardinality $N$ where $N$ is a Poisson random variable that is
independent of the sequence.
