1. Suppose $\displaystyle f: \mathbb{R} \longrightarrow \mathbb{R}$ is differentiable and that $\displaystyle f$ and $\displaystyle f'$ have no common roots. Prove or disprove that $\displaystyle f$ can have at most finitely many zeros in any interval $\displaystyle [a,b].$

(You may assume that $\displaystyle a = 0, \ b = 1.$ Note that if $\displaystyle f'$ is continuous on [a,b], then the claim is easily seen to be true. But do we really need $\displaystyle f'$ to be continuous?)


2. Suppose $\displaystyle f_n: \mathbb{R} \longrightarrow \mathbb{R}, \ n \in \mathbb{N},$ is a pointwise convergent sequence of continuous functions on the interval $\displaystyle [a,b].$ Show that $\displaystyle \{f_n \}$ is uniformly bounded on some subinterval of $\displaystyle [a,b].$

(Again you may assume that $\displaystyle a=0, \ b=1.$ Note that the subinterval doesn't have to be [a,b] itself.)