Consider a Brownian motion $B$ and let $f(r)=\sqrt{2r \ln(\ln(r))}.$ 

Is it true that $\lim_{r \to +\infty}\frac{1}{f(r)}(B_r-B_{\left \lfloor{f(r)}\right \rfloor})= 0$ a.s. ?

If so, how to prove it? Otherwise what counter-example do you suggest?