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Thread: uniformly on compact subsets

  1. #1
    Member thaopanda's Avatar
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    uniformly on compact subsets

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