Proof of Parseval's theorem in Fourier series

[Jncik]

Homework Statement

I want to prove this

[PLAIN]http://img84.imageshack.us/img84/918/asdfo.png

The Attempt at a Solution

here is part of the proof

[PLAIN]http://img824.imageshack.us/img824/4513/parseval.png

i can't understand the red part, can someone help me? thanks

edit:

i forgot to add the last part

[PLAIN]http://img847.imageshack.us/img847/4513/parseval.png

 

[micromass]

In the second step, split the integrals so that you get:

$$\frac{1}{T}\int_{-T}^T{\frac{a_0}{4}dx}+\frac{1}{T}\int_{-T}^T{\left(\sum{...}\right)^2dx}+\frac{1}{T}\int_{-T}^T{a_0\left(\sum{...}\right)dx}$$

The first integral is easy to calculate. You don't need to do anything in the second integral. Only the last integral might pose a problem. Change the sum in the last integral to

$$\sum{...}=f(x)-\frac{a_0}{2}$$

Now the third integral is also nice...