Exponential of Real Number is Strictly Positive/Proof 2

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Theorem

Let $x$ be a real number.

Let $\exp$ denote the (real) exponential function.


Then:

$\forall x \in \R : \exp x > 0$


Proof

This proof assumes the limit definition of $\exp$.

That is, let:

$\displaystyle \exp x = \lim_{n \to \infty} f_n \left({x}\right)$

where $f_n \left({x}\right) = \left({1 + \dfrac x n}\right)^n$


First, fix $x \in \R$.

Let $N = \left\lceil{\left\vert{x}\right\vert}\right\rceil$, where $\left\lceil{\cdot}\right\rceil$ denotes the ceiling function.


Then:

\(\displaystyle \exp x\) \(=\) \(\displaystyle \lim_{n \to \infty} f_n \left({x}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \to \infty} f_{n + N} \left({x}\right)\) $\quad$ Tail of Convergent Sequence $\quad$
\(\displaystyle \) \(\ge\) \(\displaystyle f_{n + N} \left({x}\right)\) $\quad$ Exponential Sequence is Eventually Increasing and Limit of Bounded Convergent Sequence is Bounded $\quad$
\(\displaystyle \) \(>\) \(\displaystyle 0\) $\quad$ Corollary to Exponential Sequence is Eventually Increasing $\quad$


$\blacksquare$