Brownian motion – Hölder continuity

Let $B$ stand for a Brownian motion on a finite interval $[0,1]$. If I am not wrong, I think that there exists a positive constant $c$, such that almost surely, for $h$ small enough , for all $0< t < 1- h$

\begin{align}
|B(t+h)-B(t)| < c\sqrt{h\log(1/h)}
\end{align}
or something like this. As a result

\begin{align}
\bigg|\frac{B(t+h)-B(t)}{h}\bigg| < K(h)
\end{align}

Am I correct ?

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