# Does the inequality used in the squeeze theorem have to hold true for all values of n?

I thought it had to be true for all values of n, but my friend is saying that "small n values are not representative of when the sequence converges so they don't have to be accounted for".

For context, the question is about proving the convergence of the sequence $$\frac{c^n}{n!}$$

The inequality my friend used with the squeeze theorem was

$$0\le \frac{c^n}{n!} \le \frac cn$$

It can be seen that if you substitute in numbers like c=3 and n=2 the inequality is false.

• Your friend is right: all you need is for there to be an $N$ such that for all $n\ge N$, the inequality holds at $n$. Mar 25, 2018 at 4:59

Due to the nature of $n!$ and $n^c$ your friend is correct, Generally through the convergence of a sequence has to be taken as the limit approaches infinity for the $a_{n}$ as $n \to \infty$