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