Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Power Series Expansion

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

As the Limit of a Sequence
implies the following definition:

Proof
Let $\exp x$ be the real function defined as the limit of the sequence:


 * $\exp x := \displaystyle \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$

From the General Binomial Theorem:

From Power over Factorial, this converges to:


 * $\exp x - \paren {\dfrac {x^0} {0!} + \dfrac {x^1} {1!} + \dfrac {x^2} {2!} + \dfrac {x^3} {3!} + \cdots} = 0$

as $n \to +\infty$.