Euler's Number: Limit of Sequence implies Base of Logarithm

Theorem
Let $e$ be Euler's number defined by:
 * $\ds e := \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n$

Then $e$ is the unique solution to the equation $\map \ln x = 1$.

That is:
 * $\map \ln x = 1 \iff x = e$

Proof
First we prove that $e$ is a solution to $\map \ln x = 1$:

From Logarithm is Strictly Increasing, $\ln$ is strictly monotone.

By Strictly Monotone Mapping with Totally Ordered Domain is Injective it follows that $\ln$ is an injection.

So the solution to $\map \ln x = 1$ is unique.

Hence the result.