into smaller pieces and do not remove any existing cuts.

If is a refinement of then

and

If P₁, P₂ are two partitions of the same interval (one need not be a refinement of the other), then

and it follows that

Riemann sums always lie between the corresponding lower and upper Darboux sums. Formally, if and together make a tagged partition

(as in the definition of the Riemann integral),