Bounded convergence theorem

Take $X = [0,1]$ with Lebesgue measure. Then let $$f_n = n 1_{[0,\frac{1}{n})}.$$ Then $f_n \rightarrow 0$ a.e. However for all $n$, $$ \int \lvert f_n - 0\rvert = \int \lvert f_n\rvert = 1$$