# asymptotic statement and one example

how I can logically understand this is false ($$c$$ is constant):

$$\frac{n!}{c} \leq 2^n$$

why it was false for large value from asymptotic notation concept? ( i see this is true for some $$n < n_0$$ but not hold for large value).

• This inequality is false for large $n$. The "asympotic notation concept" can't tell you that. You have to prove it directly (you say you can "see this.) Then you make the asymptotic statement. Just how large $n$ has to be depends on $c$. Dec 12, 2020 at 19:26
• @BettyAnderson See my answer. Dec 12, 2020 at 19:47

For each value of $$c$$ you can find out exactly where the inequality switches over by checking numerically. For example, for $$c=3$$ the change is between $$n=4$$ and $$n=5$$. There is no formula.
The argument that for any particular $$c$$ it happens eventually does not depend on knowing just where. It usually does not matter just where - that's when knowing what happens "asymptotically" is all you care about.
You have to show that for any particular $$c$$, eventually $$x_n = \frac{n!}{2^n} > c.$$ When $$n > 5$$, show that each $$x_n$$ is more than twice its predecessor. That should be enough to finish the proof.