Hmm you're right it's no more rigorous what I said. (For each positive number the expression must eventually, after finitely many terms, always be smaller than that number thereafter)
But you're not right in that the upper bound of that expression eventually stops decreasing, in fact for simple f and g as long as f is positive and eventually bounded by g the expression will decrease infinitely often if my intuition is correct
But you're not right in that the upper bound of that expression eventually stops decreasing, in fact for simple f and g as long as f is positive and eventually bounded by g the expression will decrease infinitely often if my intuition is correct