Prove that the set of endpoints of removed intervals in the Cantor middle thirds set is a dense subset of the Cantor set.
Attempt at proof:
Since each subinterval is of length $(1/3)^n$, any point contained in $K_n$ is at a distance of less than or equal to $(1/3)^n$ from an endpoint in $K_n$ for all $n$. Thus there exists a sequence of points in $K_n$ that converge to an endpoint in $K_n$.