Let $(\Omega,\mathcal{A},\mu)$ be a finite mesure space, and $\{f_n\}$ and $\{g_n\}$ are two $L^1$-bounded sequences, such that : $$ \sum_{n\geq 1}{\frac{1}{n}(F_n(f_n)(\omega)-g_n(\omega))}<\infty\qquad a.e $$ with: $F_n(f_n)=f_n1_{|f_n|\leq n}$

Can we say that: $$ \sum_{n\geq 1}{\frac{1}{n}(f_n(\omega)-g_n(\omega))}<\infty\qquad a.e $$