Exponential Function is Well-Defined/Real/Proof 2

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 sequence definition of $\exp$.

Let $\left\langle{ f_n }\right\rangle$ be the sequence of mappings $f_n : \R \to \R$ defined as:
 * $f_n \left({ x }\right) = \left({ 1 + \dfrac{x}{n} }\right)^{n}$

Fix $x \in \R$.

Then:

Thus, $\left\langle{ f_n \left({ x }\right) }\right\rangle$ is bounded above.

From Exponential Sequence is Eventually Increasing:
 * $\exists N \in \N : \left\langle{ f_{N + n} \left({ x }\right) }\right\rangle$ is increasing

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

From Tail of Convergent Sequence, $\left\langle{ f_{n} \left({ x }\right) }\right\rangle$ converges to $z$.

Hence the result, from Limit of Function Unique.