Let $f_n(x)$ be a sequence of functions defined on an closed and bounded interval of the real number system. Is it possible that each $f_n(x)$ is integrable but $f_n(x)$ doesn't converge for any $x$ in the domain ?
My thoughts are if $f_n(x)$ diverge to infinity then $f_n(x)$ may not be integrable. But what about if the sequence $f_n(x)$ is bounded but oscillatory. I can't find any clue. Also the integral may be improper one. So what is the answer then ?
Any hint will be of great help. Thank you.