Define what is meant by a nowhere dense set in a metric space. State a version of the Baire Category Theorem. Show that any complete non-empty metric space without isolated points is uncountable.

Let $A$ be the set of real numbers whose decimal expansion does not use the digit 6 . (A terminating decimal representation is used when it exists.) Show that there exists a real number which cannot be written as $a+q$ with $a \in A$ and $q \in \mathbb{Q}$.