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 $\sequence {f_n}$ 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:
| \(\ds \map {f_n} x\) | \(=\) | \(\ds \paren {1 + \dfrac x n}^n\) | Definition of $\map {f_n} x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \binom n k \frac {x^k} {n^k}\) | Binomial Theorem: Integral Index | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \frac {x^k} {k!} \frac {\paren n \times \paren {n - 1} \times \paren {n - 2} \times \cdots \paren {n - k + 1} }{n \times n \times n \times \cdots n}\) | Definition of Factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \frac {x^k} {k!} \paren 1 \paren {1 - \frac 1 n} \paren {1 - \frac 2 n} \cdots \paren {1 - \frac {k - 1} n}\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \size {\sum_{k \mathop = 0}^n \frac {x^k} {k!} \paren 1 \paren {1 - \frac 1 n} \paren {1 - \frac 2 n} \cdots \paren {1 - \frac {k - 1} n} }\) | Negative of Absolute Value | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \frac {\size x^k} {k!} \paren 1 \paren {1 - \frac 1 n} \paren {1 - \frac 2 n} \cdots \paren {1 - \frac {k - 1} n}\) | Absolute Value Function is Completely Multiplicative | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \sum_{k \mathop = 0}^n \frac {\size x^k} {k!}\) | Multiplication of Positive Number by Real Number Greater than One | |||||||||||
| \(\ds \) | \(<\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\size x^k} {k!}\) | Sum of positive terms is increasing | |||||||||||
| \(\ds \) | \(<\) | \(\ds \infty\) | Series of Power over Factorial Converges |
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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 Real Function is Unique.
$\blacksquare$