Power over Factorial

Theorem
Let $$x \in \mathbb{R}: x > 0$$ be a positive real number.

Let $$\left \langle {x_n} \right \rangle$$ be the sequence in $\mathbb{R}$ defined as $$x_n = \frac {x^n} {n!}$$.

Then $$\left \langle {x_n} \right \rangle$$ converges to zero.

Proof
We need to show that $$x_n \to 0$$ as $$n \to \infty$$.

Let $$N \in \mathbb{N}$$ be the smallest natural number which satisfies $$N > x$$.

(From the Archimedean Principle, such an $$N$$ always exists.)

First we show that $$\forall n > N: \frac {x^n} {n!} \le \frac {x^{N-1}} {\left({N-1}\right)!} \left({\frac {x} {N}}\right)^{n - N + 1}$$.