Originally Posted by

**davidciprut** he upper and lower integrations are different from each other. However he uses the same partition for both. But shouldn't there be other conditions? Such as when the partition consist of the rational points, when it consists of irrational points and when it consists of rational and irrational points. He used the same partition for U and L.

And secondly , the condition of integrability is

L={L(f,P}|P is partition for the closed interval [a,b]}

U={U{f,P}|P is partition for the closed interval [a,b]}

supL<infU (less or equal)

A function is integrable when supL=infU

So how come chosing randomly a partition prove the above?

Note: I uploaded the proof