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The Borel-Cantelli first lemma applied to that finite sum says that almost surely, for all sufficiently large $n$,For $2^{-n-1}\le t\le 2^{-n}$, both the Brownian running maximum and the positive function $\sqrt{t}f(t)$ are nondecreasing. Thus eventuallyThis proves the lower limit bound one, with interpolation across all small real times rather than only the dyadic sequence.

For every positive integer $m$, the function $mf$ satisfies the same assumptions and integrability condition. Repeating the argument gives an eventual lower bound $S_t\ge m\sqrt{t}f(t)$ on a probability-one event. Take the countable intersection of these events. On it, the ratio is eventually at least any prescribed positive integer, proving the integral lower envelope for the Brownian maximum: