I am currently reading about the mathematical construction of Brownian motion from a set of notes by Dr. P. Sousi, and I am slightly confused by the inclusion of a proof which seems to be redundant to me. For the purposes of this question, I use $B = (B_t)_{t \ge 0} $ to refer to a standard Brownian motion in $\mathbb{R}$ ($d=1$).
On page 55 (Thm. 6.14), it is proven that the hitting time of $(0,\infty)$ is almost surely $0$ i.e.
$$\mathbb{P} ( \inf \{t : B_t > 0\} = 0) = 1$$
The statement of the next proposition (Prop. 6.15), paraphrased, includes:
Let $S_t := \sup_{s \le t} B_s$. Then, for every $\varepsilon > 0$, $S_\varepsilon > 0 $ almost surely.
which is later proved directly.
Does this not follow immediately from Thm. 6.14, and if not, why not?