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 $x$.

Fix $x \in \R$.

Then:

$\ds n$

$\ge$

$\ds \ceiling {\size x}$

$\ds \leadsto \ \ $

$\ds n$

$>$

$\ds -x$

$\ds \leadsto \ \ $

$\ds 1$

$>$

$\ds \frac {-x} n$

dividing both sides by $n$

$\ds \leadsto \ \ $

$\ds 1 + \frac x n$

$>$

$\ds 0$

adding $\dfrac {-x} n$ to both sides

$\ds \leadsto \ \ $

$\ds \paren {1 + \frac x n}^n$

$>$

$\ds 0$

$\blacksquare$