$ \int_{\{|f_n|>M\}}|f_n|\leq\int_{(0,1)}|f_n-f|+\int_{\{|f_n|>M\}}|f| $
$ Since \int_{(0,1)}|f_n-f|\to0(n\to\infty), it suffices to show\sup\int_{\{|f_n|>M\}}|f|\to0(M\to\infty) $
