Proof that plane with $N$ lines can be painted with two colors so that any two neighboring regions are painted in different colors

$N$ lines are drawn in a plane. Proof that we can paint all the sectors in $2$ different colors, so any two neighbors (border's length is $>0$ (not just a point) ) are in different colors?

I am really stuck with this one. Any tips? Thank you :)

Next, suppose we've succeeded for $n$ lines. Then add more line and invert all colors on one side of it :) You can fill in the details.