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".