Limit of the sequence $\{n^n/n!\}$, is this sequence bounded, convergent and eventually monotonic?

Question

I am trying to check whether or not the sequence $$a_{n} =\left\{\frac{n^n}{n!}\right\}_{n=1}^{\infty}$$ is bounded, convergent and ultimately monotonic (there exists an $N$ such that for all $n\geq N$ the sequence is monotonically increasing or decreasing). However, I'm having a lot of trouble finding a solution that sufficiently satisfies me.

My best argument so far is as follows,

$$a_{n} = \frac{n\cdot n\cdot n\cdot \ldots\cdot n}{n(n-1)(n-2)(n-3)\dots(2)(1)} = \frac{n}{n}\cdot \frac{n}{n-1}\cdot \ldots \cdot \frac{n}2\cdot n$$

so $\lim a_{n}\rightarrow \infty$ since $n<a_{n}$ for all $n>1$. Since the sequence is divergent, it follows that the function must be ultimately monotonic.

This feels a little dubious to me, I feel like I can form a much better argument than that, or at the very least a more elegant one. I've tried to assume $\{a_{n}\}$ approaches some limit $L$ so there exists some $N$ such that

$|a_{n} - L| < \epsilon$ whenever $n>N$ and derive a contradiction, but this approach got me nowhere.

Finally, I've also tried to use the fact that $\frac{a_{n+1}}{a_n}\rightarrow e$ to help me, but I couldn't find an argument where that fact would be useful.

Answer 1

HINT for the last part: Note that

$$\frac{a_{n+1}}{a_n}=\frac{\frac{(n+1)^{n+1}}{(n+1)!}}{\frac{n^n}{n!}}=\frac{(n+1)^{n+1}n!}{n^n(n+1)!}=\frac{(n+1)^{n+1}}{n^n(n+1)}=\left(\frac{n+1}n\right)^n\;.$$

Answer 2

To show that $a_n$ is unbounded, we can show that, $a_n > n$ for all $n$. And since $n$ is unbounded by Archimedes, $a_n$ must also be unbounded.

$$a_n = \frac{n^n}{n!} = \underbrace{\frac{n}{n} \cdot \frac{n}{n-1} \cdot \frac{n}{n-2} \cdot \frac{n}{n-3}\cdots \frac{n}{2}}_{\text{Since $n \geq 1$, each of these terms are $\geq 1$}} \cdot \frac{n}{1} \geq n$$