So I would say that for b the sup is 1 and inf is 0 as you suggest.
For the sup, set m=0 and let n be anything you like, you've got 1 and nothing you can do can make it bigger.
Second, set n = 0 and let m = anything you like, you've got 0.
Are you using the convention that or ? In the first case the answers will be the same but you won't have 0 and 1 actually in the sets described.