 Pointwise supremum need not preserve Riemann integrability

Enumerate the rational points of a compact nondegenerate interval as $q_1,q_2,\dots$. Each singleton indicator function $f_n=1_{\{q_n\}}$ is Riemann integrable with integral zero. Their pointwise supremum is the Dirichlet function, which is not Riemann integrable. Thus even a uniformly bounded countable supremum need not preserve integrability.