Exponential Sequence is Eventually Strictly Positive

Theorem
Let $\sequence {E_n}$ be the sequence of real functions $E_n: \R \to \R$ defined as:
 * $\map {E_n} x = \paren {1 + \dfrac x n}^n$

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

That is:
 * $\forall x \in \R: \forall n \in \N: n \ge \ceiling {\size x} \implies \map {E_n} x > 0$

where $\ceiling x$ denotes the ceiling of $x$.

Proof
Fix $x \in \R$.

Then: