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Borel Cantelli give you the probability of $\limsup$ rather than $\lim$ to be zero, which implies that you could only find a subsequence which converges to zero $\mathbb{P}-a.s.$....
Really neat, but could we develop the problem more detailed, by which I mean does the subadditive ergodic theorem help with the Law of the iterated logarithm for infinite dimensional BM?
