Proposition: If the set of functions $\{ \phi_{1}, \phi_{2},... \} \ $ has the property that $ f = 0 $ is the only function with that all its Fourier Coefficients are zero, that is $C_{n}= 0\ $ $\ \forall n \in \mathbb{N}$. Then $\{ \phi_{1}, \phi_{2},... \} \ $ is complete
Where $$ C_{n} = \frac{ \int_{a}^{b} f\phi_{n} dx}{ \int_{a}^{b} \phi_{n}^{\ 2} dx } $$
we say that the set of functions $\{\phi_{1},\phi_{2,...} \}\ $ is complete. If $$ \lim_{N\rightarrow \infty}{ \int_{a}^{b} { \Big(f-\sum_{n=1}^{N}C_{n}\phi_{n} \Big)^2 } dx} = 0 $$
for every function $f$ with the property that ( $f$ is square-integrable. )$$ \int_{a}^{b} f^2 dx < \infty$$
I don't how any idea how to prove this.
how would you prove this? ^_^
In your definition of completeness, in the integral it should be the absolute value squared, if you also want to deal with complex-valued functions.
Suppose $f$ is a square-integrable function with all Fourier coefficients zero. Then by your definition of completeness, $$0=\lim_{N\to\infty}\int_a^b|f-\sum_{n=1}^NC_n\phi_n|^2\,dx=\int_a^b|f|^2\,dx$$ which implies that $f=0$ in $L^2(a,b)$.
It follows that the only function orthogonal to all the basis elements is the zero function.
Edit:
Following a comment by the OP. Apparently the OP takes as definition the fact that the only function orthogonal to $\phi_n$ in $L^2(a,b)$ is the zero function, and wants to prove that $\sum_{n=1}^{\infty}C_n\phi_n$ converges to $f$ for every $f\in L^2(a,b)$, where $C_n=(f,\phi_n)/\|\phi_n\|^2$. Note that as of now, 6.8.2018, 6:13 a.m GMT, the question suggest exactly the opposite direction. Anyway, the two being equivalent, here is the other direction:
Assume that the only function orthogonal to $\phi_n$ in $L^2(a,b)$ is the zero function. We want to prove that if $C_n=(f,\phi_n)/\|\phi_n\|^2$, then $$\sum_{n=1}^{\infty}C_n\phi_n=f,$$ which is equivalent to $$\lim_{N\to\infty}\|\sum_{n=1}^NC_n\phi_n-f\|=0$$ We are assuming that $\{\phi_n\}_{n=1}^{\infty}$ is an orthogonal system, not yet complete. Then it follows from Bessel's inequality that the series $\sum_{n=1}^{\infty}C_n\phi_n$ converges in $L^2(a,b)$. The only thing left is to show that it converges to $f$ itself. To show this, consider the function
$$g=f-\sum_{n=1}^{\infty}C_n\phi_n$$ It is easy to check that $(g,\phi_m)=0$ for all $m$, and so by hypothesis, $g$ is the zero function, and we are done.