I've got a proof of Lebesgue Dominated Convergence theorem, but it is long and unwieldy--and I have exams coming up soon so I was trying to understand the proof on the Wikipedia page.
However, as usual I'm stuck on something they claim is "trivially" true, which means I'm just missing something basic.
They state, that as $f_n\to f$ pointwise. Then $\limsup_{n\to \infty}|f-f_n|=0$. Why is this?
For example, what if $f_n(x)=\bigg\{\begin{split}1, &\quad 0\leq x\leq 1/n\\ 0,&\quad 1/n<x\leq 1\end{split}\;\;$ on the interval $[0,1]$
Then the function converges pointwise to zero, but wouldn't $\limsup_{n\to\infty}|f-f_n|=1$? I get that this would be true if we had uniform convergence, but is there some reason we can say this for pointwise that I'm missing?
Is it implicit that we're ignoring sets of measure zero? Are we using some fact as these functions are dominated by the integrable function $g$? What am I missing?
If I can figure out the justification for this point, then the rest of the proof is golden and much cleaner than what I was doing before, but since it hinges on this fact, I want to make sure it's valid, and at the moment I don't see it.
Thanks for any help.