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Let $f:[0,1] \rightarrow[0, \infty)$ be a continuous function. Let $$ a=\inf _{0 \leq x \leq 1} f(x) \text { and } b=\sup _{0 \leq x \leq 1} f(x) . $$ For every positive integer $m$, define $$ c_{m}=\left[\int_{0}^{1}(f(x))^{m} d x\right]^{1 / m} . $$ Prove that $c_{m} \in[a, b]$, for all $m \geq 1$, $\displaystyle{}\lim _{m \rightarrow \infty} c_{m}$ exists and find its value.
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