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

Theorem
Let $e$ be Euler's number defined by:
 * $\displaystyle e := \lim_{n \to \infty} \left({ 1 + \frac{1}{n} }\right) ^n$

Then $e$ is the unique solution to the equation $\ln \left({ x }\right) = 1$.

That is:
 * $\ln \left({ x }\right) = 1 \iff x = e$

Proof
First we prove that $e$ is a solution to $\ln \left({ x }\right) = 1$:

Now, since $\ln$ is strictly increasing, it is  strictly monotone.

And since $\ln$ is strictly montone, it is injective. (Strictly Monotone Mapping with Totally Ordered Domain is Injective)

So the solution to $\ln \left({ x }\right) = 1$ is unique.

Hence the result.