How can I prove the following equation?
$\{\omega:f_n(\omega)\nrightarrow f(\omega)\} = \bigcup_{k=1}^\infty\bigcap_{N=1}^\infty\bigcup_{n=N}^\infty\{\omega: |f_n(\omega)-f(\omega)|\geq \frac{1}{k}\}$ . I was thinking that maybe $RHS = \{\omega: |f_n-f|>0, i.o.\}$, but to be honest I'm not sure about what that limsup may exactly be referred to.
RHS is the set of the points $\omega$ such that there is some $k\ge1$ such that for all $N$ there is some $n\ge N$ such that $\lvert f_n(\omega)-f(\omega)\rvert\ge\frac1k$. Which is in fact the negation of the statement $\lim_{n\to\infty}f_n(\omega)= f(\omega)$.