Let $(X_n)_{n\in\mathbb N}$ be independent real random variables, with values in $(0,\infty)$. Consider the random variable $Z(w):=\inf\limits_{n\in\mathbb N}X_n(w)$. Prove that for every fixed $\epsilon>0$ the following inclusion holds.
$$\{Z=0\}\subset\limsup\limits_{n}\{X_n<\epsilon\}$$
My Attempt:
$\displaystyle\{Z=0\}=\{w:\inf_nX_n=0\}=\bigcap\limits_{n\ge 1}\bigcup\limits_{k\ge 1}\{X_k\le\frac1n\}\subset\bigcap\limits_n^{\lceil\frac{1}{\epsilon}\rceil}\bigcup\limits_{k\ge 1}\{X_k\le\frac1n\}\subset\limsup\limits_{n}\{X_n<\epsilon\}$
Is that correct ?
Define $A_s:=\bigcup_{k\geqslant 1}\{X_k\leqslant s\}$. Since $A_s\subset A_t$ if $s\lt t$, we have $$\bigcap_{n=1}^{\varepsilon^{-1}}A_{1/n}=A_\varepsilon$$ which is not contained in $\limsup_n\{ X_n\leqslant\varepsilon\}$.
However, we can notice that the equality $$\{Z=0\}=\bigcap_{n\geqslant 1}\bigcap_{j\geqslant 1}\bigcup_{l\geqslant j}\left\{X_l\leqslant \frac 1n\right\}$$ holds because $X_k(\omega)$ is positive for any $\omega$. Then we reduce the intersection over $n\geqslant 1$ to $n\leqslant 1/\varepsilon$.