Proving a theorem about Fourier coefficients I need to prove this:

Let $f$ be a $C^1$ function on $[-\pi, \pi]$. Prove that the Fourier coefficients of $f$ satisfy $|a_n| \leq \frac{K}{n}$ for some constant $K$.

Can someone please let me know if I would be on a right track if I said:
Let $||f(x)||_\infty$ = C, then $a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \cos(nx) dx \leq \frac{2}{\pi} C \int_{0}^{\pi} \cos(nx) dx = \frac{2}{\pi} C \frac{\sin(nx)}{n} \leq C \frac{2}{\pi}\frac{1}{n} = \frac{K}{n}$.
Any suggestions would be greatly appreciated.

More easily you can do this:
$a_n(f)=-\frac {b_n(f')}{n}$ and by Riemann-Lebesgue lemma we have that $b_n(f')\to 0$ and thus $|b_n(f')|\leq K$ for every $n$ and  we have that $|a_n| \leq \frac{K}{n}$. 
