how would one show that the interior of the cantor set is the empty set?
You can't because the cantor set is nonempty compact set of measure zero.
I will not show this but... let E be the cantor set then
$\displaystyle x \in [0,1] \mbox{ and } x \in E \mbox{ if and only if } x $
has a ternary expansion whose digits
$\displaystyle b_k \ne 1 \mbox{ }\forall \mbox{ }k $