Let $B_t$ be a Brownian motion in $\left[0,1\right]$. I am pretty sure that is is possible to prove that, for all $\varepsilon>0$ there exists an $0<h<\varepsilon$ such that
$$ \left|B_{t+h}-B_t\right|\geq C\,\sqrt{h\,\log\left(1/h\right)}, \text{ a.s.} $$
for all $t\in[0,1-h]$. There is a proof in the book by Morters and Peres (2010) (Theorem 1.13) which is not pretty clear to me, even the statement says that "there exist $0<h<\varepsilon$ and $t\in[0,1-h]$":
while I think it should be for all $h\in\left(0,\varepsilon\right)$ and $t\in[0,1-h]$. In fact in the proof they claim to have proved that, for all $\varepsilon>0$
Which is pretty strange to me, I would expect something as
$$ \forall h\in\left(0,\varepsilon\right),\forall t\in[0,1-h],\mathbb{P}\left\{\left|B_{t+h}-B_t\right|\leq c\,\sqrt{h\,\log(1/h)} \right\}=0 $$
Am I missing something? Any suggestion?