Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Extension of Rational Exponential

Theorem
The following definition of the concept of the real exponential function:

As the Limit of a Sequence
implies the following definition:

Proof
Let the restriction of the exponential function to the rationals be defined as:


 * $\ds \exp \restriction_\Q: x \mapsto \lim_{n \mathop \to +\infty}\left ({1 + \frac x n}\right)^n$

Thus, let $e$ be Euler's Number defined as:


 * $e = \ds \lim_{n \mathop \to +\infty}\left ({1 + \frac 1 n}\right)^n$

For $x = 0$:

For $x \ne 0$:

where the continuity in the last step follows from Power Function to Rational Power permits Unique Continuous Extension.

For $x \in \R \setminus \Q$, we invoke Power Function to Rational Power permits Unique Continuous Extension.