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.

$\blacksquare$