What's the relationship between lim sups of random variables and lim sups of events?

Question

I'm reading David Williams' Probability with Martingales.

On page 41 he performs the following manipulation

$$ L := \lim \sup(X_n/\log n)\\ \implies \mathbb{P}(L \geq 1) \geq \mathbb{P}(X_n > \log n,\text{ infinitely often}) $$

It appears to me as if by some sleight of hand $\lim \sup$ is being used in two very different ways here. The first $\lim \sup$ is one of a function, presumably pointwise. The second $\lim \sup$, which appears in the form of "infinitely often" is one of sets.
Fundamentally, what I think it boils down to is how the following two example sets, written out in all detail, relate to another.

$$ \left\{\omega : \lim_{m \rightarrow \infty} \sup_{n \geq m} X_n(\omega) \leq x\right\} $$

$$ \bigcap_m \bigcup_{n \geq m} \{\omega : X_n(\omega) \leq x\} $$

I can sorta see that the first is contained in the second. But I'm not convinced that the second is contained in the first.

Answer 1

The implication has a $\geq$ in it. If $X_n>\log n$ infinitely often for a fixed $\omega$, that means there's a sequence $n_i$ with $X_{n_i}(\omega)>\log n_i$. This implies $\limsup_n X_n(\omega)/\log_{n_i}>1$ by definition of limsup. Hence the $\geq$ follows.

Answer 2

Alternatively, think of an event (such as $E_n=[X_n>\log n]$ as a 0-1 valued function, the indicator of the event: $\mathbf 1_{E_n}(\omega)=1 \iff X_n(\omega)>\log n$, and so on. The definition of $\limsup$ for events was chosen so that this kind of punning works out correctly.