Preferences exhibiting local nonsatiation
Since we are working in $\mathbb{R}_+$ you can't go below (0,0). So consider the point (0,0). Are we guaranteed a point around it that is preferred? So using the notation you used, let $(x_1,x_2)=(0,0)$, then any other point $(y_1,y_2)$ where $y_1$ or $y_2$ is positive will be worse than $(0,0)$. So we have a satiation point at $(0,0)$. 

Now consider anywhere else. We can also go lower by reducing either coordinate so there is always a better point until we reach both points =0.