limits - Why is this sum wrong?

$$\lim_{n\rightarrow\infty}\sum_{r=1}^n \sin\left(\frac{r}{n} \right)=\lim_{n\rightarrow\infty} \left[\sin\frac{1}{n}+\sin\frac{2}{n}+\cdots+\sin(1)\right]=0+0+\cdots+\sin(1)=1$$

Could anybody explain why this is wrong? I've tried to see why this doesn't work but I don't see why not. Thank you.