Please give me a hint on the following question.
Given $x \in [0, 1]$ show that there exist $a_1, a_2, a_3,...\in S_m$ such that for $s_n=\sum_{k=1}^n \frac{a_k}{m^k}$ we have $x=\lim_{n\rightarrow \infty} s_n$.
I'm not good at proving existence questions.