Exponential of Real Number is Strictly Positive/Proof 5/Lemma
Theorem
Let $x$ be a real number.
Let $\exp$ denote the (real) Exponential Function.
Then:
- $\forall x \in \R: \exp x \ne 0$
Proof
This proof assumes the definition of $\exp$ as the solution to an initial value problem.
That is, suppose $\exp$ satisfies:
- $(1): \quad D_x \exp x = \exp x$
- $(2): \quad \map \exp 0 = 1$
on $\R$.
Aiming for a contradiction, suppose $\exists \alpha \in \R: \exp \alpha = 0$.
Suppose that $\alpha > 0$.
Let $J = \closedint 0 \alpha$.
From Exponential Function is Continuous, $\exp$ is continuous on $J$.
From Max and Min of Function on Closed Real Interval:
- $\exists K \in \R: \forall x \in J: \size {\exp x} < K$
Then, $\forall n \in \N : \exists c_n \in J$ such that:
\(\ds 1\) | \(=\) | \(\ds \exp 0\) | by $(2)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 0}^{n - 1} \frac {\map {\exp^j} \alpha} {j!} \paren {-\alpha}^j + \frac {\exp c_n} {n!} \paren {-\alpha}^n\) | Taylor's Theorem for Univariate Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 0}^{n - 1} \frac {\map \exp \alpha} {j!} \paren {-\alpha}^j + \frac {\exp c_n} {n!} \paren {-\alpha}^n\) | by $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 0}^{n - 1} \frac 0 {j!} \paren {-\alpha}^j + \frac {\exp c_n} {n!} \paren {-\alpha}^n\) | from our assumption aiming for a contradiction | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\exp c_n} {n!} \paren {-\alpha}^n\) |
So:
- $\forall n \in \N: 1 \le K \dfrac {\alpha^n} {n!}$
That is, dividing both sides by $K$:
- $\forall n \in \N: \dfrac 1 K \le \dfrac {\alpha^n} {n!}$
But from Power over Factorial, $\dfrac {\alpha^n} {n!} \to 0$.
This contradicts our assumption.
The same argument, mutatis mutandis proves the result for $\alpha < 0$.
By hypothesis $(2)$:
\(\ds \alpha\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exp \alpha\) | \(=\) | \(\ds 1\) | |||||||||||
\(\ds \) | \(\ne\) | \(\ds 0\) |
Hence the result.
$\blacksquare$