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:

$\displaystyle \exp x = \lim_{n \mathop \to \infty} \map {f_n} x$

where $\map {f_n} x = \paren {1 + \dfrac x n}^n$


First, fix $x \in \R$.

Let $N = \ceiling {\size x}$, where $\ceiling {\, \cdot \,}$ denotes the ceiling function.


Then:

\(\ds \exp x\) \(=\) \(\ds \lim_{n \mathop \to \infty} \map {f_n} x\)
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \map {f_{n + N} } x\) Tail of Convergent Sequence
\(\ds \) \(\ge\) \(\ds \map {f_{n + N} } x\) Exponential Sequence is Eventually Increasing and Limit of Bounded Convergent Sequence is Bounded
\(\ds \) \(>\) \(\ds 0\) Corollary to Exponential Sequence is Eventually Increasing


$\blacksquare$