Exponential of Sum/Real Numbers/Lemma

Lemma
Let $x, y, n \in \R$ where $n \ne -\left({x + y}\right)$.

Let $n \in \N_{> 0}$ such that $n > -\left({x + y}\right)$.

Then:


 * $\displaystyle 1 + \frac{x + y} n + \frac{xy}{n^2} = \left(1 + \frac{x + y} n \right)\left({1 + \frac{\left({\frac{xy}{n+x+y} }\right)} n} \right)$

Proof
As $n \in \N_{> 0}$ we have that $n \ne 0$ and so the fractions in the expressions are defined.

That final step is justified, as we have that $n > -\left({x + y}\right)$ and so $n + x + y \ne 0$.

Also see
This lemma is used in the proof of Exponent of Sum:
 * $\exp \left({x + y}\right) = \left({\exp x}\right) \left({\exp y}\right)$

for real $x$ and $y$.

Note that this result is actually valid for all $n \in \R$ subject to the given conditions, but in the context in which it is used, $n \in \N$ is a large natural number tending to infinity.