  In 1984, K. Mahler asked how well elements in the Cantor middle third set can
be approximated by rational numbers from that set, and by rational numbers
outside of that set. We consider more general missing digit sets $C$ and
construct numbers in $C$ that are arbitrarily well approximable by rationals in
$C$, but badly approximable by rationals outside of $C$. More precisely, we
construct them so that all but finitely many of their convergents lie in $C$.
