Let m be the Lebesgue measure on $[0, 1]$. Suppose that $\{f_n\}$ is a sequence of Borel measurable functions on $[0, 1]$ that converges almost everywhere to $f$. For each $n, k$ let $$E_{n,k}= \bigcup_{m\ge n} \Big\{x : |f_m(x)-f(x)|>\frac{1}{k}\Big\}$$ Show that $\lim_{n\to \infty}m(E_{n,k})=0$ for each $k$.
Is the following correct? for each $k$ \begin{align} E_k =\lim_{n\to \infty} E_{n,k}= \bigcap_{n=1}^\infty \bigcup_{m\ge n}E_{n,k} =\bigcap_{n=1}^\infty\bigcup_{m\ge n}\Big\{x : |f_m(x)-f(x)|>\frac{1}{k}\Big\} \end{align} where $E_k$ is decreasing in $k$ (???????), but since $f_n \to f$ a.e.
$$m(\bigcup_{k\ge 1}\bigcap_{n=1}^\infty \bigcup_{m\ge n}E_{n,k})=0$$ $$\bigcap_{n=1}^\infty \bigcup_{m\ge n}E_{n,k}\subset \bigcup_{k\ge 1}\bigcap_{n=1}^\infty \bigcup_{m\ge n}E_{n,k} \implies \lim_{n\to \infty}m(E_{n,k})=0$$