User blog comment:Edwin Shade/Understanding The Infinite/@comment-25601061-20171109021425/@comment-31663462-20171109235023

Imagine this:

$$ (\omega + 3) \cdot n \le \omega^2, ~ n < \omega $$

As $$ n \to \omega $$, it is clear that the limit will be $$ \ge \omega \cdot \omega = \omega^2 $$, but since we know that $$ (\omega + 3) \cdot n \le \omega^2 $$, the limit must come out to give

$$ (\omega+3) \cdot \omega = \omega^2 $$