Please show that for all $x,y\in\mathbb{R}$,
$$e^{x+y} - e^xe^y = \lim_{k\to\infty} \sum_{n=1}^k \sum_{j = 0}^n\left(\frac{x^{k+j}}{(k+j)!}\frac{y^{n-j}}{(n-j)!} + \frac{x^j}{j!}\frac{y^{k+n-j}}{(k+n-j)!}\right)\;.$$
(This is from a homework assignment)
I'm thinking of a way to proceed by induction, but it is not clear to me how to utilize it for this problem. Can someone work out this problem so I can ask questions from it?