A Neat Problem - Points on a sphere

I came across a little riddle while studying for the Putnam Mathematics Contest. I thought it was neat, so I'm passing it along.

Suppose there is a sphere on which you draw five tiny dots. Prove that there is a way to cut the sphere in half (simply into hemispheres - no zig-zag cutting allowed!) so that four of the dots are on the same hemisphere, regardless of where the dots are placed. You can count dots "on the line" of the cut toward whichever hemisphere you wish.

If you have questions which begin with "Is it ok if I...," the answer is very likely "no."

Enjoy!