real analysis – Estimating the sine of the angles between vectors

Let ${s_n}$ be a sequence of unit vectors in $mathbb{R}^2$ converging to some $sinmathbb{R}^2$. I want to show that $$sinmeasuredangle(s_n,s)le sum_{m=n}^inftysinmeasuredangle(s_m,s_{m+1})$$ for all $n$. This estimate makes intuitive sense from drawing a picture, but I am struggling to find a formal argument. Any suggestions would be appreciated.