Exponential Function is Well-Defined/Real/Proof 2

Proof
This proof assumes the sequence definition of $\exp$.

Let $\sequence {f_n}$ be the sequence of mappings $f_n : \R \to \R$ defined as:
 * $\map {f_n} x = \paren {1 + \dfrac x n}^n$

Fix $x \in \R$.

Then:

Thus, $\sequence {\map {f_n} x}$ is bounded above.

From Exponential Sequence is Eventually Increasing:
 * $\exists N \in \N: \sequence {\map {f_{N + n} } x}$ is increasing

From Monotone Convergence Theorem (Real Analysis), $\sequence {\map {f_{N + n} } x}$ converges to some $z \in \R$.

From Tail of Convergent Sequence, $\sequence {\map {f_n} x}$ converges to $z$.

Hence the result, from Limit of Function Unique.