Pointwise supremum need not preserve Riemann integrability
= Pointwise supremum need not preserve Riemann integrability
Enumerate the rational points of a compact nondegenerate interval as $q_1,q_2,\ldots$. 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.