prove that $\lim_{n\rightarrow \infty} S_NF(\frac{1}{\pi n})=\int_{0}^{1}\frac{\sin(\frac{t}{\pi})}{2t}dt$

Question

I've been trying to prove that $\displaystyle\lim_{n\rightarrow \infty} S_NF(\frac{1}{\pi n})=\int_{0}^{1}\frac{\sin(\frac{t}{\pi})}{2t}dt$

this was in a quiz 2 years ago. I assume it somehow is related to Darboux sums, the problem I face is that I can't understand how a Fourier series of a constant isn't a constant. When I take the limit of the expression in the SNF I get a zero. Does anyone understand?