I am trying to prove if f and g are Riemann integrable, then fg is also Riemann integrable using Lebesgue's integrability criterion. I already proved that a Riemann integrable function is bounded. Not much harder to show fg is too bounded. How do I show that [a,b] is of measure zero? I can't figure outa sequence whose infinite union contains [a,b] but also whose sum of lengths is less than all epsilon greater than zero. How do a construct such a sequence? Or should I go about it differently?