Question

suppose f is an integral element function on [0,1], p and q are partitions of [0,1].

for any partitions A and B, let P = A U B. Then Up-Lp <= Ua-La.

is this statement true or false? either provide a proof or counterexample.

cross "element"

Answer 1

Note : A U B = P implies we have more points in P than A. So for points x1 and x2 in A we can have x0 in between where x0 is in P. Thus adding more points decrease the upper sum as is clear from the above figures. And the opposite case holds for Lower sum. That is Increasing points in the partition makes the lower sum larger.