Questions on the Problem of Induction

1. Despite the argument he presents 'against' induction, Hume does not suggest that we stop using it. Why not?

1'. A related question is: when is it rational to believe in things that can't be proved?

2. The falsificationist rejects induction as essential to the process of scientific discovery: a logic of induction is not required to produce good theories, we only need normal deductive logic to eliminate bad theories. But is this really enough? Does this mean that whenever we are faced with the problem of improving a theory we have to sift through an infinite number of randomly generated possible theories in order to find those that aren't immediately falsified? Obviously not. but why not?

3. Does it make any sense to talk about one false theory being 'closer to the truth' than another false theory? (Use examples.)