Question as in the image
note: I have been following Postol's Mathematical Analysis. I used the def to get integral formula for $$c_k=\int_0^{2\pi}f(x)\cdot e^{-ikx}$$, and got another formula of $$c_k=e^{-i\pi}\int_{-\frac{\pi}{k}}^{2\pi-\frac{\pi}{k}}f(t+\frac{\pi}{k})\cdot e^{-ikt}$$ using the change of variable hint. Now, I am supposed to get the average of these two formulas, and somehow apply the above Lipschitz condition to prove. (I got stuck at this step)
Hint: $$\int_{-\frac{\pi}{k}}^{2\pi-\frac{\pi}{k}}f(\frac{\pi}{k})\cdot e^{-ikt}=0$$ so $$c_k=e^{-i\pi}\int_{-\frac{\pi}{k}}^{2\pi-\frac{\pi}{k}}[f(t+\frac{\pi} k)-f(\frac {\pi} {k})]\cdot e^{-ikt}.$$ Now apply Lipschitz condition and integrate.