Computing the value of a function based on the geometric series of 1/2^n

Question

So in John B. Conway's First Course in Analysis Book we're told to consider the interval $[a,b],$ and let $\{r_n\}$ denote a sequence of all rationals in said interval. Next, we define a map $\alpha:[a,b]\to\mathbb{R}$ by $$\alpha(t)=\sum_{\substack{n\in\mathbb{N}\\r_n<t}}\dfrac{1}{2^n}.$$ Totally fine with that definition. Where I'm confused is when he says that $\alpha(b)=1.$ If $b$ is irrational surely that would be the case as $\alpha(b)$ would just be a geometric sum which converges to $1$. If $b$ is rational, however, it seems like $\alpha(b)$ would never be $1$. Consider $r_n=b$ for some $n\in\mathbb{N}$, hence $\alpha(b)=1-1/2^n<1.$ So I'm confused as to why Conway says it's $1$. Could anyone explain? Thank you.