Say $x \in [0, 1)$ and $\displaystyle x = \sum_{i = 1}^\infty \frac{a_i}{10^i}$ for $a_i \in \{0, 1\}$. Is it true if $\displaystyle x = \sum_{i = 1}^\infty \frac{b_i}{10^i}$ for $b_i \in \{0, 1\}$ that $a_i = b_i$ for all $i$?
It seems to me that yes this is true, since the only times I have seen uniqueness of decimal representation break down is when we introduce repeating 9s. Is there a simple proof of my statement above?
EDIT: I added base 10 to the subject to clarify I don't want to consider $\displaystyle \sum_{i = 1}^\infty \frac{a_i}{2^i}$. I am not sure if that is the correct terminology.
I want to establish $f : \{0, 1\}^\infty \to \mathbb{R}$ by $(a_1, a_2, a_3, \cdots) \mapsto 0.a_1a_2a_3\cdots$ is an injection.