There are many primes which have digits only from the set of one digit primes $\mathcal S_p = \{2,3,5,7\}$, but can we prove that there exists at least one $p$ for each $n$: $10^n \leq p \leq10^{n+1}$, $n\in \mathbb N$ with digits only in $\mathcal S_p$?
some first few examples $$7, 23, 257, 2357, \cdots$$
It feels like something easy to verify for many $n$ but difficult to prove for all.
EDIT This is not the same question as the duplicate marker at all. It asks about primes of all sizes, not only three digit and only asks about solely being built up by single-digit primes.