Exponential Function is Well-Defined/Real/Proof 2

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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:

\(\displaystyle f_n \left({ x }\right)\) \(=\) \(\displaystyle \left({ 1 + \dfrac x n }\right)^n\) Definition of $f_n \left({ x }\right)$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^n {n \choose k} \frac{x^k}{n^k}\) Binomial Theorem: Integral Index
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^n \frac{x^k}{k!} \frac{ \left({ n }\right) \times \left({ n - 1 }\right) \times \left({ n - 2 }\right) \times \cdots \left({ n - k + 1 }\right) }{ n \times n \times n \times \cdots n }\) Definition of factorial
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^n \frac{x^k}{k!} \left({ 1 }\right) \left({ 1 - \frac 1 n }\right) \left({ 1 - \frac 2 n }\right) \cdots \left({ 1 - \frac{k - 1} n }\right)\)
\(\displaystyle \) \(\le\) \(\displaystyle \left\vert{ \sum_{k \mathop = 0}^n \frac{x^k}{k!} \left({ 1 }\right) \left({ 1 - \frac 1 n }\right) \left({ 1 - \frac 2 n }\right) \cdots \left({ 1 - \frac{k - 1} n }\right) }\right\vert\) Negative of Absolute Value
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^n \frac{ \left\vert{ x }\right\vert^k }{ k! } \left({ 1 }\right) \left({ 1 - \frac 1 n }\right) \left({ 1 - \frac 2 n }\right) \cdots \left({ 1 - \frac{k - 1} n }\right)\) Absolute Value Function is Completely Multiplicative
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{k \mathop = 0}^n \frac{ \left\vert{ x }\right\vert^k }{ k! }\) Multiplication of Positive Number by Real Number Greater than One
\(\displaystyle \) \(<\) \(\displaystyle \sum_{k \mathop = 0}^\infty \frac{ \left\vert{ x }\right\vert^k }{ k! }\) Sum of positive terms is increasing
\(\displaystyle \) \(<\) \(\displaystyle \infty\) Series of Power over Factorial Converges

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.

$\blacksquare$