It does not approach it from above or below. As N goes to infinity, the proportion of n<N for which An==”Left” need not converge to 1⁄2, but it must have 1⁄2 as a limit point, so the proportion of n<N for which An==”Left” is arbitrarily close to 1⁄2 infinitely often. Further, the same is true for any easy to compute subsequence of rounds.
So, unfortunately it might be that An goes left many many times in a row e.g. for all n between 1010 and 10100, but it will still be unpredictable, just not locally independent.
It does not approach it from above or below. As N goes to infinity, the proportion of n<N for which An==”Left” need not converge to 1⁄2, but it must have 1⁄2 as a limit point, so the proportion of n<N for which An==”Left” is arbitrarily close to 1⁄2 infinitely often. Further, the same is true for any easy to compute subsequence of rounds.
So, unfortunately it might be that An goes left many many times in a row e.g. for all n between 1010 and 10100, but it will still be unpredictable, just not locally independent.