Let $E$ denote the set of real numbers in $[0,1]$ with digits $1$ and $3$ only in their development in base $5$. How to prove that $E$ is dense in $[0,1]$?
Is this the right way to see that E is dense?: $\forall x\in [0,1],\forall \epsilon\gt0,\exists n_0\in \mathbb N \mid d(x,e_n)\lt\epsilon,\forall n \ge n_0$ and $(e_n)_n\subset E$.
If so, how to find such a sequence $(e_n)_n$?
The set is clearly not dense in $[0,1]$: for example, $0$ is neither a limit point nor a point of $E$.
You may be wanting to prove that $E$ is nowhere-dense in $[0,1]$. That is, we say a set $E$ is nowhere-dense if its closure has an empty interior.