Let \(S\) be the set of sequences \[ a=(a_1,a_2,\ldots), \qquad a_k\in\{0,1\}. \] Show that the map \(\theta:S\to\mathbb R\) defined by \[ \theta(a)=\frac{a_1}{10}+\frac{a_2}{10^2}+\cdots \] is injective. Include an explanation of why the infinite series converges.
Hint: if \(a\ne b\), you may suppose their first differing coordinate is \(n\) and write \[ a=(a_1,\ldots,a_{n-1},0,a_{n+1},\ldots), \qquad b=(a_1,\ldots,a_{n-1},1,b_{n+1},\ldots). \]