TITLE: Formalizing composition of sequences
QUESTION [6 upvotes]: The following first equality is intuitively evident:
$$
\lim_{n\to \infty} \left(1+\frac{x+y+\frac{xy}{n}}{n}\right)^n 
= \lim_{n\to \infty} \left(1+\frac{x+y}{n}\right)^n
=e^{x+y}
$$
how can I give a formal argument?
Informally it is something like
$$
\lim_{n\to \infty} \left(1+\frac{x+y+\frac{xy}{n}}{n}\right)^n 
=\lim_{n\to \infty} \left(1+\frac{\lim_{n\to \infty}\left(x+y+\frac{xy}{n}\right)}{n}\right)^n 
=\lim_{n\to \infty} \left(1+\frac{x+y}{n}\right)^n 
$$
Is there a "classical" property of sequences that backs this argument? Something like limit of composition, but here we speak of sequence only:
$$
(b_n),\quad b_n \to b\quad\text{ for }n\to\infty \implies 
\lim_{n\to \infty} \left(1+\frac{b_n}{n}\right)^n
=\lim_{n\to \infty} \left(1+\frac{b}{n}\right)^n
$$
EDIT As Qiaochu Yuan is important to define what I have already defined.
This computation is at the beginning of "Real Analysis", lets say in the chapter of sequences. Later, in the notes of complex analysis, I develop formal series with derivation and everything is the standard way (I think).
By this question I tried another approach, very early (1) define
$$
e^x = \lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n
$$
(2) derive the product as depicted in this question
(3) derive the derivation of the $e^x$.
Thus I have (a) no logarithm, (b) no derivation of $e^x$ yet.

REPLY [3 votes]: Another approach: Note that
$$\left(1+\frac{x+y+\frac{xy}{n}}{n}\right)^n 
= \left(1+\frac x n\right)^n\left(1+\frac y n\right)^n.$$
Addendum
We have that
\begin{align*}
\left(\frac{1+\frac{x+y+\frac{xy}{n}}{n}}{1+\frac{x+y}{n}}\right)^n
&= \left(
\frac{\left(1+\frac x n\right)\left(1+\frac y n\right)}{1+\frac{x+y}{n}}\right)^n 
\longrightarrow \frac{e^x e^y}{e^{x+y}} \quad(n\to\infty).
\end{align*}
It remains to show that this limit is unity.
See this answer, for example.