Fourier's Theorem
An arbitrary periodic function $F(t)$ with period $T$ can be decomposed into a linear combination of the functions $f_n(t)$ and $g_n(t)$ where, $$f_n(t)=\sin \frac {2πnt}{T}$$ $$g_n(t)=\cos \frac {2πnt}{T}$$ Mathematically, $$F(t)=b_0 + b_1g_1(t) +b_2g_2(t) + …… + a_1f_1(t) +a_2f_2(t) + ……,$$ where $n$ is a non-negative integer and all of $a_i$,$b_i$ are real.
Problem
As $F(t)$ is periodic with period $T$ , $$F(t+nT)=F(t)…….… (1)$$. What I want is that I can derive the RHS directly from equation (1). However, I am unable to do so, perhaps in lack of theoretical knowledge. Any suggestions are welcome.
Note
What ever proofs I have got as yet, they assume the decomposition for some complex exponential and justify it using some arguments. I don't want anything like that