Let X=[0,1] , A=$\displaystyle [0,1]\cap B$. (B is the Borel-$\displaystyle \sigma$ algebra), $\displaystyle \lambda$ the lebesgue measure. In $\displaystyle (X,A,\lambda)$ explain why these statements are true or false:

1) if $\displaystyle \exists 1 \le p < \infty$ with $\displaystyle f_n \in L_p$ and $\displaystyle ||f_n||_p \rightarrow 0$ then $\displaystyle ||f_n||_\infty \rightarrow 0$

2) if $\displaystyle f_n \in L_p$ and $\displaystyle ||f_n||_p \rightarrow 0 \forall 1 \le p < \infty$ then $\displaystyle ||f_n||_\infty \rightarrow 0$

Well I honestly do not have much ideas how to solve this. It's clear 1) isn't true because 2) wouldn't make sense if 1) was true. But a formal explanation? A prove? I don't know.

All I tried was finding functions that are for example in L_1 but not in L_2 . But I couldn't find a contradiction. Also there exist Lebesgue-integrable functions that are not bounded...??

Could please someone give me a hint?