## A Couple of Nice Problems - Analysis

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

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

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

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