uniformly on compact subsets

**thaopanda** · November 15th 2009, 12:34 PM

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.

**Jose27** · November 15th 2009, 01:44 PM

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 $n> \frac{ \ln ( \epsilon )}{ \ln (x_0)}$

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