Exponential Function is Well-Defined/Real/Proof 4

Theorem
Let $x \in \R$ be a real number.

Let $\exp x$ be the exponential of $x$.

Then $\exp x$ is well-defined.

Proof
This proof assumes the Inverse of $\ln$ definition of $\exp$.

Let $k \in \N$.

Let $J_k = \left[{ \dfrac{1}{k}, \,.\,.\, k }\right]$

Let $J_{k}^{o} = \left({ \dfrac{1}{k}, \,.\,.\, k }\right)$

From Natural Logarithm Function is Continuous, $\ln$ is continuous on $J_k$.

Also:

From Derivative of Inverse Function, $\ln$ is invertible.

Hence the result, from Inverse Mapping is Unique.