As f is increasing over the interval [a,b] we can state that f(x+δx)>f(x), where δx=(b-a)/n.
Ln takes two consecutive values f(x+δx) and f(x) and uses the rectangular area f(x)δx, while Rn uses the rectangular area f(x+δx)δx, therefore f(x+δx)δx>f(x)δx. It follows then that Rn>Ln when the areas are summed over the interval. As n increases δx→0, therefore f(x+δx)δx→f(x)δx, implying that Rn decreases as Ln increases.