It will give up eventually if you are really persistant. You could try
carefully hand-applying Roundup (or its equivalent) to the sprouts as
they come up.

When the roots of my neighbor's cottonwood (which was severely butchered
in a pruning) started doing that in my yard, heavy doses of Roundup did
nothing to them. I finally got sick, and just pulled the roots up all the
way back to the tree - the roots which were producing the suckers were small
and shallow, it wasn't that hard to do - and it took care of the problem.

steve