Q:

Proving that:$\lim\limits_{n \to \infty }\int |f_n(x)|\log(1+ |f_n(x)|)dm=0$

Let $f_n(x)$ be a sequence in $L_2(X; m) $ such that $$ \lim_{n \to \infty }\| f_n\|_2=0 $$ 
To prove $$\lim_{n \to \infty }\int |f_n(x)|\log(1+ |f_n(x)|)dm=0$$ I want to use dominated convergence theorem, but it's hard to find a g such that  $$|f_n(x)|\log(1+ |f_n(x)|\leq g$$ for any n.

A:

well,  for all we have $x>0$ $$\log(x+1) =\int_0^x\frac{dt}{t+1}\le x$$
Therefore, 
$$\lim_{n \to \infty }\int |f_n(x)|\log(1+ |f_n(x)|)dm \le \lim_{n \to \infty }\int |f_n(x)|^2dm =\lim_{n \to \infty }\|f_n\|^2_2=0$$

