I am working on the billiards problem and need a proof for the following:

A ball hit at a $45$ angle from the corner of a rectangular table will always eventually go into a pocket no matter what the dimensions of the table are.

For this problem, we are not considering friction, spinning the ball, or dimensions that do not consist of whole numbers. The pockets are only located in the four corners. I have found that the ball always reaches a pocket for dimensions $1$x$1,1$x$2,2$x$2,1$x$3,2$x$3,$ all the way through $7$x$7$ by drawing the shapes and angles of incidence and reflection, but I do not know how to write a proof for this. Thanks!