For each n $\displaystyle \in$ N, let $\displaystyle F_{n}$ : (0,1) $\displaystyle \rightarrow$ R be given by $\displaystyle F_{n}(x)$ := $\displaystyle x^nsin(\frac{1}{x^{n-1}})$. Show that $\displaystyle F_{n} \rightarrow $ F as n $\displaystyle \rightarrow \infty$ uniformly on compact subsets of (0,1) where F $\displaystyle \equiv$ 0.