# dominated convergence theorem application

Prove or disprove this statement: if $f_n : \mathbb R \to \mathbb R$ are integrable functions with $f_n \to 0$ pointwise and $|f_n(x)| \le \frac{1}{|x| + 1}$ for all $n$, $x$, then $\lim\limits_{n \to \infty} \int\limits_{\mathbb R} f_n = 0$.

According to Dominated Convergence theorem $\lim\limits_{n \to \infty}\int\limits_{\mathbb R} f_n = \int\limits_{\mathbb R} f$.

I don't know how one can show that it is equal to 0 or not. I am stuck...need some help

• According to the stated convergence of $f_n$, what is $f$? – Alex S Jun 24 '15 at 17:08
• How are you applying the Dominated Convergence Theorem? What integrable function dominates all $f_n$? – breeden Jun 24 '15 at 17:17
• it is the function g(x)=1/(|x|+1) – user250109 Jun 24 '15 at 17:20
• @Anna that function is not integrable. – breeden Jun 24 '15 at 17:23
• does this mean that I can not apply dominated convergence theorem? @breeden – user250109 Jun 24 '15 at 17:26

Consider the functions $$f_n(x) = \frac{\chi_{[n,n^2]}(x)}{1+x}.$$
Hint: What happens if $$f_n(x) = \frac{n}{ (|x| +n)^2 }$$
• you should find that $\int f_n dx = 2$ for all $n$. Thus this is a counter example to the statement. – Jeb Jun 24 '15 at 17:43