Convergence in integral norm

Question

Let $f_n$ be a bounded sequence in $L^1$ norm that converges pointwise to $f\in L^1$. I have to show that $$||f_n||_1-||f_n-f||_1\to ||f||_1$$ I tried to use the Lebesgue theorem but I cannot guarantee that there is a function in $L^1$ that bounds all $f_n$ pointwise. I cannot use Peppo Levi theorem since there is no monothone convergence and I tried with Fatou's lemma without getting anything useful.

Answer 1

The reversed triangle inequality give us $$ ||f_n(x)|-|f_n(x)-f(x)||\leqslant |f_n(x)-f_n(x)+f(x)|=|f(x)| $$ Now apply the dominated convergence theorem.