title: Result of multiplying both sides of not-big-O relationship by the same function?
body: <p>Is the following generally true?
  <br><code>f∉O(g) ⇒ f*h∉O(g*h)</code><br>
  Where f, h, g are positive-only functions. My intuition is that it is true, but I don't know how to prove it. </p>
  
  <p>Here is why I think it is true:<br>Because f∉O(g) there is no c multiplied by g that will make f ≤ c*g for a large enough x. There is no valid c for g because either f and g alternate who is on top or f dominates g. For each x, the point f(x) and g(x) will be scaled by h(x), therefore if f(x) is above g(x), then h(x)f(x) will be above h(x)g(x). This would mean the lack of asymptotic dominance must remain the same when multiplied by h, right?</p>