An increasing sequence $(x_i)^n_{i=1}$ of positive integers is an $n$-term Egyptian underapproximation of $\theta\in (0, 1]$ if $\sum^n_{i=1} {1\over x_i} < \theta$. A greedy algorithm constructs an $n$-term underapproximation of $\theta$. For some but not all numbers $\theta$, the greedy algorithm gives a unique best $n$-term underapproximation for all $n \geq 1$. An infinite set of rational numbers is constructed for which the greedy underapproximations are best, and numbers for which the greedy algorithm is not best are also studied.
Mel Nathanson