Let $(H, \|\cdot\|)$ denote a Hilbert space which is continuously embedded in $L^2(\mathbb R^n)$. Let $\{f_n\}$ be a sequence such that
$$ f_n\to f \text{ a.e. in } \mathbb R^n,$$and$$\limsup_{n\to +\infty} \|f_n\|\le K,$$where $K>0$ is a constant.
As an exercise for my calc class, I have to decide if these conditions are enough to show that $\|f\|$ is bounded. If yes, I have to calculate if $\|f\|$ is bounded by $K$ or a different constant.
According to me, one has$$\|f\|\le \liminf_{n\to +\infty} \|f_n\|\le \limsup_{n\to +\infty} \|f_n\|\le K,$$and then the answer is yes.
I am not totally sure about my argument, because of the almost everywhere convergence. Is it enough to argue as I did?Is my argument correct?
