For each $n = 1, 2, 3, . . .,$ let $f_n : \Bbb R → \Bbb R$ be the function $f_n = \chi[n,n+1)− \chi[n+1,n+2)$; i.e., let $f_n(x)$ equal $+1$ when $x ∈ [n, n + 1)$, equal $−1$ when $x ∈ [n + 1, n + 2)$, and $0$ everywhere else. Show that $\int_\Bbb R \sum_{n=1}^\infty f_n \ne \sum_{n=1}^\infty\int_\Bbb R f_n$
My work: Notice that $f_n+f_{n+1}= χ[n,n+1)− χ[n+2,n+3)$. This implies that $\sum_{n=1}^{\infty} f_n= \chi[1,2)$ because every other point the characteristic function become zero for all rest of the partial sum. Therefore, $\int_\Bbb R \sum_{n=1}^\infty f_n =\int_\Bbb R \chi[1,2)=1 $ but $\sum_{n=1}^\infty\int_\Bbb R f_n=0$. Thus we are done. Did I miss anything?
I was wondering if you could help me to solve this problem. I appreciate your kind help. Thank you!