Let be a meromorphic function on the complex plane, and suppose there is an integer such that has at most points for all . Show that is a rational function.

The back of the book says:

Choose such that the number of points in is maximum. Then attains values near only near points in , is bounded at , and is meromorphic on hence rational. This seems a bit too sketchy. I think that is bounded at by taking . How is meromorphic on ? Also, how does attain values near only near points in . I understand that it is the maximum. However, I don't see precisely why. Thanks.