1. Suppose is differentiable and that and have no common roots. Prove or disprove that can have at most finitely many zeros in any interval
(You may assume that Note that if is continuous on [a,b], then the claim is easily seen to be true. But do we really need to be continuous?)
2. Suppose is a pointwise convergent sequence of continuous functions on the interval Show that is uniformly bounded on some subinterval of
(Again you may assume that Note that the subinterval doesn't have to be [a,b] itself.)