  An increasing sequence $(x_i)_{i=1}^n$ of positive integers is an $n$-term
Egyptian underapproximation of $\theta \in (0,1]$ if $\sum_{i=1}^n
\frac{1}{x_i} < \theta$. 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 $n$.
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.
