A partition of an interval [a,b] is a finite sequence
Each [xi − 1,xi] is called a subinterval of the partition. A refinement of the partition
is a partition
such that for every i with
there is an integer r(i) such that
xi = y ...
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
.
It follows that
Riemann sums always lie between the corresponding lower and upper Darboux sums. Formally, if
< ...
