Exponential Function is Well-Defined/Real/Proof 1
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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 power series definition of $\exp$.
From Series of Power over Factorial Converges:
- $\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$ converges
Hence the result, from Convergent Real Sequence has Unique Limit.
$\blacksquare$