Exponential of Sum/Real Numbers/Proof 5

Theorem

Let $x, y \in \R$ be real numbers.

Let $\exp x$ be the exponential of $x$.

Then:

$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$

This proof assumes the definition of $\exp$ as a series.

Then:

$\ds \map \exp {x + y}$

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {x + y}^n$

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac 1 {n!} \sum_{k \mathop = 0}^n \frac {n!} {k! \paren {n - k}!} x^k y^{n - k}$

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \sum_{k \mathop = 0}^n \paren {\frac 1 {k!} x^k} \paren {\frac 1 {\paren {n - k}!} y^{n - k} }$

$\ds $

$=$

$\ds \paren {\sum_{n \mathop = 0}^\infty \frac {x^n} {n!} } \paren {\sum_{n \mathop = 0}^\infty \frac {y^n} {n!} }$

Definition of Cauchy Product

$\ds $

$=$

$\ds \map \exp x \, \map \exp y$

$\blacksquare$