Proposition: Size of Riemann Sums

Suppose P = { x₀, x₁, x₂, ..., x_n} is a partition of the closed interval [a, b], f a bounded function defined on that interval. Then we have:

• The lower sum is increasing with respect to refinements of partitions, i.e.
  L(f, P) L(f, P') for every refinement P' of the partition P
• The upper sum is decreasing with respect to refinements of partitions, i.e.
  U(f, P) U(f,P') for every refinement P' of the partition P
• L(f, P) R(f, P) U(f, P) for every partition P