# 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:

 $\displaystyle n$ $\ge$ $\displaystyle \left \lceil{\left \vert{x}\right \vert}\right \rceil$ $\displaystyle \implies \ \$ $\displaystyle n$ $>$ $\displaystyle -x$ Real Number is between Ceiling Functions and Negative of Absolute Value $\displaystyle \implies \ \$ $\displaystyle 1$ $>$ $\displaystyle \frac{-x} n$ dividing both sides by $n$ $\displaystyle \implies \ \$ $\displaystyle 1 + \frac x n$ $>$ $\displaystyle 0$ adding $\dfrac{-x} n$ to both sides $\displaystyle \implies \ \$ $\displaystyle \left({1 + \frac x n}\right)^n$ $>$ $\displaystyle 0$ Power of Strictly Positive Real Number is Strictly Positive: Positive Integer

$\blacksquare$