Why is epsilon not a rational number?

See Surreal number :

Consider the smallest positive number in $S_ω$:

$\varepsilon =\{S_{-}\cup S_{0}|S_{+}\}=\{0|1,{\tfrac {1}{2}},{\tfrac {1}{4}},{\tfrac {1}{8}},...\}=\{0|y\in S_{*}:y>0\}$.

This number is larger than zero but less than all positive dyadic fractions. It is therefore an infinitesimal number, often labeled $ε$.

Thus epsilon is, "by definition" less than (and so different from) all rational in the $(0,1)$ interval.