Let be an integralable function on , and let be periodic with period . Prove
for all .
Proof. For any with we know from a previous exercise that there exists a unique such that
Then, we start by splitting the integral into two pieces. (The goal here is to rearrange the integral so that it starts at and ends at , and then use the fact that is periodic to conclude that this integral is the same as the one from 0 to .)
This completes the proof