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▲ dataflow 6 hours ago

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".