Prove that $\frac{n^n}{3^n} < n! < \frac{n^n}{2^n} \forall n \geq 6 $

Prove that $$\frac{n^n}{3^n} < n! < \frac{n^n}{2^n} \forall n \geq 6 $$

I'm trying induction, this is what I have so far:

Basecase $(n=6): 64<720<729$ is true.

Inductive Case: Assume the inequality $\frac{n^n}{3^n} < n! < \frac{n^n}{2^n} $ holds $ \forall n $ such that 6 $\leq n \leq k $, want to show that the inequality will hold for $n=k+1$.

What next?

From $\exp(x)=\sum_{n=0}^\infty \frac{x^n}{n!}> \frac{x^n}{n!}$ for $x>0$ you get immediately that $\exp(n)> \frac{n^n}{n!}$. Knowing that $e<3$ we conclude $3^n>\exp(n)>\frac{n^n}{n!}$, hence the left inequality.

When the right hand side grows from $\frac{n^n}{2^n}$ to $\frac{(n+1)^{n+1}}{2^{n+1}}$ this is by a factor of $$ \frac{(n+1)^{n+1}}{2n^n}=\frac{n+1}2 \frac{(n+1)^{n}}{n^{n}}=\frac{n+1}2 \left(1+\frac1n\right)^{n}$$ The term $\left(1+\frac1n\right)^n$ should be well-known to converge monotonically $\to e$ from below. Especially, $\frac{n+1}2 \left(1+\frac1n\right)^{n}>n+1$, i.e. the right hand side grows faster than the middle term.

Hint: You have two things to prove: one is $\frac {(k+1)^{(k+1)}}{3^{(k+1)}}\lt (k+1)!$. Try to write these terms in terms of what you know about $k$, so the right side is clearly $(k+1)k!$. We can then say $(k+1)k!\gt (k+1)\frac {k^k}{3^k}$. You need to relate $(k+1)^{(k+1)}$ and $k^k$, so need to say something about $(1+\frac 1k)^k.$ Does that ring a bell?

Given:

$$\frac{n^n}{3^n} < n! < \frac{n^n}{2^n} \forall n \geq 6$$

Using Sterling's formula we want to prove the following 2 inequalities:

$$\frac{n^n}{3^n} < \sqrt{2\pi n}\left ( \frac{n}{e} \right )^n ---> [1] $$

and

$$\frac{n^n}{2^n}>\sqrt{2\pi n}\left ( \frac{n}{e} \right )^n ---> [2] $$

For the first inequality suppose that the opposite is true; that is, suppose that:

$$\frac{n^n}{3^n} \geq \sqrt{2\pi n}\left ( \frac{n}{e} \right )^n ---> [3] $$

so, $$n^n \geq \sqrt{2\pi n}\left ( \frac{3*n}{e} \right )^n$$

since $3/e$ is about $1.10363832351$ we'd have:

so, $$n^n \geq \sqrt{2\pi n}\left ( 1.01*n \right )^n$$

The above is clearly not true for any value of $n \geq 6$. Hence the proposition in [3] is false and [1] is true.

The same idea can be used to prove iniquality [2]. Note that $2/e$ is approximately 0.73575888234.