Proof that ${\bf 1}_{\mathbb{Q}}(x) $ (indicator function of the set of rational numbers) is not Riemann-Integrable over [0,1]

Question

We just saw the Riemann-Lebesgue theorem which says that a function is Riemann-integrable if the set of discontinuities has Null size.

The indicator function being defined by

I deduce that the function is equal to 0 over the real axis except for a infinite (but countable) amount of points in which it is equal to 1.

Since the size of a set of countable points is Null (see This Question), I would deduce that $$I\!I_{\mathbb{Q}}(x) $$ Is indeed Integrable - where is my mistake?