Question: Let $\{f_n \}_{n=1}^\infty$ be a sequence of non-negative real-valued continuous functions on $(0,1) \subset\mathbb R$ and such that for any continuous $\phi: (0,1) \rightarrow \mathbb R$ with compact support, the limit $$ \lim_{n\rightarrow \infty} \int \phi f_n $$ exists.
Then, is it true that for any $\phi: (0,1) \rightarrow \mathbb R$ with compact support, the limit $$ \lim_{n\rightarrow \infty} \int \phi \sqrt{1+(f_n)^2} $$ exists?
Attempt I tried triangle inequality, rewriting the integral, but just can't seem to get it. I think if the $f_n$ converge in measure that might be enough. It looks like it should be true but maybe it's not?