 Solution

True. The negative-level hitting times from the preceding calculation are almost surely finite for every negative integer, and occur at unbounded times. ThereforeThis is the almost-sure negation in the original PDF, not another assertion of convergence. The stronger oscillation statement $\mathrm{liminf}{B}_t=-\infty$ and $\mathrm{limsup}{B}_t=+\infty$ rules out that convergence path by path.