induction problem

I've been having some problems with this problem.. I just don't know where to start.

Problem:
Consider a "red-black" coloring of a set of regions to be coloring each region either red or black such that no two regions sharing the same side have the same color. For example a red and black chess board. Prove by induction that if you have a set of n infinite straight lines, the regions they produce have a red-black coloring.

I start with the base case of n = 1 and thus that one line would divide the region in half and black would go in one, red in the other. Thats about as far as I can get.

Any pointers to where to start would be great.

Okay, draw any shape and draw any number of lines through it. Label the divisions red and black as appropriate. Our hypothesis holds, right? Anyway we assume out hypothesis for n straight lines though a shape.

Now draw a straight line anywhere though the shape. So now we have our n+1 lines through and we have to show that it hold for n+1 lines too.

You notice that a the shape is divided into two by the new line. Don't change the color of any of the divisions on one side and change the color of all the divisions on the other side. Now you have n+1 lines with no sides having the same color. Fin.

So that's a way of finding a red-black coloring.... but how does one do it in mathematical induction. I don't think drawing some shapes and then saying that by flipping the colors we get our anwswer would be considered a proof.

>>2's answer is basically correct.  Just replace "shape" with "infinite plane".

You already proved it for N = 1.

Assume there is some set of N >= 1 lines in the plane and that the regions they produce are already colored accordingly.

Add any infinite line.  Color according to what >>2 suggests.  Each of the 2 partitions created by the line are colored correctly, and inverting the colors for one of them doesn't change that.  It should also be clear that no 2 regions that have this line as the border separating them have the same color.