How can we tell the relationship between two lim-sup sets?

Question

Let $f_n(x)$ be sequence of functions, and $\epsilon>0$. Denote two sets as follows $$E(\epsilon) = \limsup_{n\rightarrow\infty}\{x:|f_n(x)| >\epsilon \}$$ $$F = \limsup_{n\rightarrow\infty}\{x:|f_n(x)| > 1/n \}.$$ Based on the definitions above, can we conclude $E(\epsilon) \subset F$ for any $\epsilon>0$ ?

Answer 1

Yes.

If $x \in E(\varepsilon)$, this means that for infinitely many $n \in \Bbb{N}$ we have $|f_n(x)| > \varepsilon$ so there exists an increasing sequence $(p(n))_n$ in $\Bbb{N}$ such that $|f_{p(n)}(x)| > \varepsilon$ for all $n \in \Bbb{N}$.

Pick $n_0 \in \Bbb{N}$ such that $\frac1{n_0} < \varepsilon$. Then for all $n \in \Bbb{N}$ such that $p(n) \ge n_0$ we have $$|f_{p(n)}(x)| > \varepsilon > \frac1{n_0} \ge \frac1{p(n)}$$

and the set $\{p(n) : n \in \Bbb{N}, p(n) \ge n_0\}$ is still an infinite subset of $\Bbb{N}$ so $x \in F$.