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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

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    $\begingroup$ According to the stated convergence of $f_n$, what is $f$? $\endgroup$ – Alex S Jun 24 '15 at 17:08
  • $\begingroup$ How are you applying the Dominated Convergence Theorem? What integrable function dominates all $f_n$? $\endgroup$ – breeden Jun 24 '15 at 17:17
  • $\begingroup$ it is the function g(x)=1/(|x|+1) $\endgroup$ – user250109 Jun 24 '15 at 17:20
  • $\begingroup$ @Anna that function is not integrable. $\endgroup$ – breeden Jun 24 '15 at 17:23
  • $\begingroup$ does this mean that I can not apply dominated convergence theorem? @breeden $\endgroup$ – user250109 Jun 24 '15 at 17:26
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Consider the functions $$f_n(x) = \frac{\chi_{[n,n^2]}(x)}{1+x}.$$

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Hint: What happens if $$ f_n(x) = \frac{n}{ (|x| +n)^2 } $$

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  • $\begingroup$ can I use link ? $\endgroup$ – user250109 Jun 24 '15 at 17:32
  • $\begingroup$ calculate the integral... $\endgroup$ – Jeb Jun 24 '15 at 17:32
  • $\begingroup$ after taking the limit of integral is it equal to zero? $\endgroup$ – user250109 Jun 24 '15 at 17:43
  • $\begingroup$ you should find that $\int f_n dx = 2$ for all $n$. Thus this is a counter example to the statement. $\endgroup$ – Jeb Jun 24 '15 at 17:43
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    $\begingroup$ could you help me with calculating the integral please? $\endgroup$ – user250109 Jun 24 '15 at 17:45

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