In my notes, $$S_{xy} = \sum_{i=1}^n x_iy_i -\frac 1 n \left( \sum_{i=1}^n x_i \sum_{i=1}^n y_i \right)$$
And also the sample covariance $s_{xy} = \frac 1 {n-1}S_{xy}$
But he also rewrites it in this form elsewhere
$$s_{xy} = \frac{\sum_{i=1}^n x_iy_i} n -\overline{x}\overline{y}$$
How are the two forms for $s_{xy}$ equivalent? I can't seem to prove it.
They definitely are not equvalent: $$ s_{xy} = \frac{\sum_{i=1}^n{x_iy_i}}{n}-\overline{x}\,\overline{y} = \dfrac{S_{xy}}{n}\neq \dfrac{S_{xy}}{n-1}. $$