Formal Poisson summation formula

Question

I want to prove the following equality: $$ \lim_{a\to -\infty,b\to \infty}\sum_{n=a}^b \frac {\sin \pi (c+n)}{\pi (c+n)}=1,\text{ for any }c\in \mathbb R. $$ All solutions I found directs me to an application of Poisson summation formula applied to the function $1_{(-1,1)}$. However, one sufficient condition for Poisson summation formula to hold is that both $f$ and $\hat f$ are $O(|x|^{-1-\epsilon})$ for some $\epsilon>0$, but now $\sin (x)/x$ does not have this decay.