Yes, the statement is true.

This is a Big-O notation analysis of the factorial function $n!$, and it aligns with Stirling's approximation, which estimates the growth of $n!$ as $n$ becomes large. The approximation of $n!$ is:

$n! \sim \sqrt{2\pi n} \left( \frac{n}{e} \right)^n$

For asymptotic behavior, we focus on the dominant term $\left( \frac{n}{e} \right)^n$. Thus, the factorial $n!$ grows at a rate very close to this expression, with additional terms being lower order.

The inequality:

$n! \leq C \cdot \left( \frac{n}{e} \right)^n$

is an expression of the Big-O notation, where $C$ is some constant, and $O\left( \left( \frac{n}{e} \right)^n \right)$ means that $n!$ does not grow faster than some constant multiple of $\left( \frac{n}{e} \right)^n$.

Would you like more details, or do you have any questions?

Here are 5 related questions that expand on this topic:

1. How does Stirling's approximation help in simplifying factorial calculations?
2. What is the significance of Big-O notation in algorithmic analysis?
3. Can you derive the constant $C$ in the Big-O bound of $n!$?
4. How do we handle the lower-order terms in Stirling's approximation?
5. What are some practical applications of asymptotic approximations like $n!$ in computer science?

Tip: Big-O notation describes the upper bound of the growth rate of a function, making it particularly useful for analyzing algorithms' worst-case performance.