# Pointwise convergence a simple fonction [closed]

How can I prove that $$f_n$$ converges pointwise to $$f$$ such that

$$f_n=\begin{cases} f,&\text{ in }A_n\\ 0,&\text{ in }A_n^c \end{cases}$$ Where $$A_n=\{ x\in X\,:\, 1/n \leq f \leq n, n\in\Bbb \}$$ and $$f$$ is positive and in $$L^p$$?

## closed as off-topic by Kavi Rama Murthy, RRL, Shailesh, Ak19, Adrian KeisterJun 19 at 15:15

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$$\|f_n-f\|^p_p=\int_{A_n}|f_n-f|^p d \mu+\int_{A_n^c}|f_n-f|^p d \mu=0+\int_{A_n^c}|f|^p d \mu.$$ Now split the last integral into two cases $$f>n$$ and $$0. Think, one of the integrals is over a set of measure zero (which one and why?)