for s belongs to T

for s belongs to S-T

With this definition the open balls

are disjoint, the authors state. That is the hint.

I see from the metric that the balls are indeed disjoint. I have not been able to use this property to show that S is finite if there is a countably dense subset of functions in B(S), nor given that the set S is finite that there is a countably dense set in B(S). I surmise that perhaps the zeros of a polynomial with real polynomial of degree N is involved in showing that S is finite. I have not been able to carry through the argument, sadly.

Can anyone help?