Sequence such that $E_{n} \to E$ and $m(\cup E_{n}) = \infty$ then $\lim_{n} m(E_{n}) > m(\lim_{n} E_{n})$ (Example)

Question

I proved that

If $(E_{n})$ is a sequence of measurable sets such that $E_{n} \to E$ and $m(\cup E_{n}) < \infty$, then $\lim_{n} m(E_{n}) = m(\lim_{n} E_{n})$.

Now, I'm trying to find a example when $E_{n} \to E$ and $m(\cup E_{n}) = \infty$ such that $\lim_{n} m(E_{n}) > m(\lim_{n} E_{n})$ but I didn't succeed. Can anybody help me?

Answer 1

$E_n=[n,\infty)$ is an obvious example.