Maybe an easier explanation: just subtract g(x) from both sides. You get: f(x) - g(x) ≤ O(1) Now, if you already know that f(x) - g(x) = O(1) means "f

Maybe an easier explanation: just subtract g(x) from both sides. You get: f(x) - g(x) ≤ O(1) Now, if you already know that f(x) - g(x) = O(1) means "f and g eventually differ by no more than a constant", then f(x) - g(x) ≤ O(1) must mean "f eventually stops exceeding g by a constant".