Let f be a function on the interval [a, b] such that for every epsilon > 0 there exist integrable functions g, h on [a, b] with h(x) <= f(x) <= g(x) and ((integral from a to b g(x)dx) - (integral from a to b h(x)dx)) < epsilon. Show that f is integrable on [a, b].

Any ideas on where to start and/or how I would go about showing this? Thanks.