If $f_n + g_n \to h$ in $L^2(\Omega)$ and $f_n, g_n \geq 0$, does $f_n \to f$ for some $f$?

Question

On a bounded domain $\Omega$, if $f_n + g_n \to h$ in $L^2(\Omega)$ and each $f_n, g_n \geq 0$, does $f_n \to f$ for some $f$?

I feel like this should be true since each sequence is non-negative, so at least in a pointwise a.e. sense it ought to hold. Does the convergence hold in $L^2$ too, or do I need something more like DCT (eg. an $L^\infty$ bound) to make it work?

Answer 1

It's not true : Take $f_n = 1+ (-1)^n$ and $g_n = 1-(-1)^n$

Answer 2

No, simply choose any kind of nonnegative, bounded non-convergent sequence $f_n$ and then set $g_n = C - f_n$ for a sufficiently large constant $C$.