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**Markeur** Let $\displaystyle E$ be a Lebesgue measurable subset of $\displaystyle \mathbb{R}$ with $\displaystyle m(E) < \infty$ and let $\displaystyle (f_n)$ be a sequence of real-valued Lebesgue measurable functions on $\displaystyle E$. Prove or disprove that there exists a sequence $\displaystyle (\alpha_n)$ of positive real numbers such that $\displaystyle lim_{n \to \infty} \alpha_n f_n = 0$ almost everywhere on $\displaystyle E$.

I suppose the statement is true? So far can't think of any counterexample. If it's true, how do I go about starting the proof?

Thanks in advance.