Unable to understand the time-shifting property of CTFS
Even though Latex is highly encouraged instead of just copying stuff, in this case it would have been interesting to see the original &quot;proof&quot; because it's hard to know which errors come from the original proof and which are just typos introduced by entering the Latex code.
Of course $dt$ should have been replaced, that's just a simple typo. And the integration bounds could have been changed, but that's not necessary in this case because the integrand is $T$-periodic, and hence, as long as you integrate over one period the exact upper and lower bounds don't matter:
$$\int_{0}^Tg(t)dt=\int_{t_0}^{T+t_0}g(t)dt\qquad\text{if}\quad g(t)=g(t+T)$$