数学分析问题

设$f\in C[a,b]$.对任意的$x\in(a,b)$有$\left| f(x)\right|<1$.证明:$\lim\limits_{n\to∞}\int_{a}^{b}({f(x)})^{n}dx=0$.
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Math001

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对正数$\epsilon>0$(不妨$\epsilon<\dfrac{b-a}{2}$)

有,$|f(x)|$在$[a+\epsilon/3,b-\epsilon/3]$上有最大值$M(\epsilon)<1$

于是存在正整数$N$,当$n>N$时,有$|f(x)|^n\le (M(\epsilon))^n<\dfrac{ \epsilon}{3(b-a)}$

得到$|\int_a^b(f(x))^ndx|\le \int_a^b|f(x)|^ndx$

$=\int_{a}^{a+\epsilon/3}|f(x)|^ndx+\int_{a+\epsilon/3}^{b-\epsilon/3}|f(x)|^ndx+\int_{b-\epsilon/3}^{b}|f(x)|^ndx$

$\le \int_{a}^{a+\epsilon/3}dx+\int_{a}^{b}|f(x)|^ndx+\int_{b-\epsilon/3}^{b}dx$

$<\epsilon/3+\int_{a}^{b}\dfrac{\epsilon}{3(b-a)} dx+\epsilon/3=\epsilon$

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