(We also need the condition that is integrable. Can you think of an example when this fails?)

I start you off.

1)Assume is integrable and -periodic.

2)We will show the forward direction, i.e. is -periodic implies .

3)Say that is a -periodic function.

4)By definition it means .

5)Now #4 is true for all , pick .

6)Thus,

7)By definition of we see that .

8)Thus, .

9)Since is -periodic we use the theorem that .

10)By #9 we see that .

Q.E.D.