Suppose $I = [0, 1]$ and $f_n: I \to \mathbb{R}$ is Lebesgue measurable for all $n \in \mathbb{N}$ and $$ \int_{I} |f_n|^2 \,d\mu \leq 5 $$ for all $n\in \mathbb{N}$. Suppose moreover that $f_n(x) \to 0$ as $n \to \infty$, for every $x \in [0, 1]$. Does it follow that $\lim_{n \to \infty} \int_I |f_n|^2 \,dm = 0$? How about $\lim_{n \to \infty} \int_I |f_n| \,dm = 0$?
My intuition want to say yes to the first part of the question and no to the second part of the question. My first intuition is to use dominated convergence since we have somewhat of a bound given. However, this bound $\int_I |f_n|^2 \,dm \leq 5$ seems to be too weak for me to find a uniform bound out of it for the sequence $\{ |f_n|^2 \}$. In particular, a bounded integral does not seem to have any strong implication on the bounds of functions. I am completely lost on this one so any hint would be appreciated.
(I am a new student preparing for the preliminary exams at my school by myself and have been solving old exam problems, which are not be graded in any ways, so you might have seen me posting quite often recently. Please let me know if this is not adhering to the community guideline and I will stop posting. Thank you in advance.)