Exponential Sequence is Eventually Strictly Positive

Theorem
Let $\left \langle{E_n}\right \rangle$ be the sequence of real functions $E_n: \R \to \R$ defined as:
 * $E_n \left({x}\right) = \left({1 + \dfrac x n}\right)^n$

Then, for each $x \in \R$ and for sufficiently large $n \in \N$, $E_n \left({x}\right)$ is positive.

That is:
 * $\forall x \in \R: \forall n \in \N: n \ge \left \lceil{\left \vert{x}\right \vert}\right \rceil \implies E_n \left({x}\right) > 0$

where $\left \lceil{x}\right \rceil$ denotes the ceiling of $x$.

Proof
Fix $x \in \R$.

Then: