Let be $\alpha, \beta \in \mathbb R$ such that $\alpha < \beta $ and $x \in [\alpha, \beta ]$. Consider the random time
$$T_x = \inf \{ t\geq 0 : x+ B_t \notin [\alpha, \beta]\},$$ where $B=(B_t)_{t\geq 0}$ is a standard brownian motion in $\mathbb R$ starting from zero.
To show that $T_x < + \infty \ \mathbb P -\text{a.e.}$, one wrote the line
\begin{align}\mathbb P \left( T_x = +\infty \right) &=\mathbb P \left(\{ \forall t \geq 0, \alpha \leq x+B_t \leq \beta\} \right) \\&\leq \mathbb P \left(\{0\leq \liminf _{t \rightarrow \infty}\frac {B_t} {\sqrt{2t\log(log(t))}} \} \cup \{ \limsup _{t \rightarrow \infty}\frac {B_t} {\sqrt{2t\log(log(t)) }}\leq 0 \} \right) = 0 \end{align}
where the last equality comes by the law of the iterated logarithm.
How to justify the inequality ? Ideally by writting some intermediate line(s).
I would appreciate any advise.
Thanks in advance.