# Math Help - uniformly on compact subsets

1. ## uniformly on compact subsets

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

2. Let $K \subset (0,1)$ compact then $x^n$ attains a maximum, say at $x_0$, in $K$ and $\vert f(x) \vert \leq \vert x^n \vert \leq \vert x_0 ^n \vert < \epsilon$ as long as $n> \frac{ \ln ( \epsilon )}{ \ln (x_0)}$