We clearly must have $ ||f|| \geq 0 $.
Since $ f_n \geq 0 $, by Fatou we obtain
$ 1 \geq \liminf ||f_n|| \geq ||f|| $.
To show these bounds cannot be improved, let
$ f_{odd} = n\chi_{(0,\frac{1}{n})}, f_{even} = 3n\chi_{(0,\frac{1}{n})}, f=0 $
$ g_{odd} = 1, g_{even} = 1 + 2n\chi_{(0,\frac{1}{n})}, g=1 $
