Monotone sum of a monotone sequence with another sequence, can we get any convergence properties?

Question

Let $f_n$ and $g_n$ be functions defined on a bounded domain $\Omega$. We know that $$f_n + g_n \to h$$ pointwise a.e. (in fact it decreases to $h$), in $L^2$ and weakly in $H^1$. Furthermore, a.e., $$f_{n+1} + g_{n+1} \leq f_n + g_n$$ $$f_{n+1} \leq f_n$$ $$f_n + g_n \leq 0$$ for all $n$.

Is it possible to deduce any sort of monotonicity or convergence for the $g_n$ from this information, or any sort of norm convergence for the $f_n$?

Answer 1

The sequences $(f_n)$ and $(g_n)$ need not be bounded: Set $f_n(x)=-n$, $g_n=-f_n$. Then $f_n+g_n$ converges, all the other properties are satisfied, but $(f_n)$ and $(g_n)$ are unbounded.