Let E$\displaystyle \subset$N a infinite subset. Show that exist a$\displaystyle \in$R such that {floor(a^k); k $\displaystyle \in$N}$\displaystyle \cap$E, contains infinites integers.

I try this:

floor(a^k) =n$\displaystyle \in$E$\displaystyle \Longleftrightarrow$k log a$\displaystyle \in$(log n, log n+1) and ?????nothing