__Claim:__ If f(x) is a REAL-valued function on x E [-L,L], then the full Fourier series is exactly equivalent to the complex Fourier series.

This is a claim stated in my textbook, but without any proof. I also searched some other textbooks, but still I have no luck of finding the proof.

I've already spent an hour thinking about how to show that this is true, but still I am not having much progress. Here is what I've got so far:

Full Fourier series is:

where

Complex Fourier series is:

where

And now I am having trouble with this...how can I use the last part to show that if f(x) is REAL-valued, the complex Fourier series can be reduced to the full Fourier series. Can someone please show me how to continue from here? I also don't see how a sum from negative infinity to infinity (for complex Fourier series) can possibly be reduced to a sum from 0 to infinity (for full Fourier series). It seems like I have no hope...

I am really frustrated now and any help is very much appreciated!

[note: also under discussion in s.o.s. math cyberboard]