Let $\displaystyle f :X \rightarrow R$ be a measurable function in $\displaystyle X$ with $\displaystyle m(X)< \infty$.

Show that if $\displaystyle f^n$ is Lebesgue integrable for every $\displaystyle n$ and that $\displaystyle \lim_{n\to\infty}\int f^n\,dm$ exists in $\displaystyle R$,then $\displaystyle -1\le f(x)\le 1$ almost every where.

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