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

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

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.

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

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

Hence the result.