Q:

Proof of existence of $e^x$ and its properties

I have been looking at the proof of the existence of $e^x$ and its properties, and I understand the induction argument which yields the Taylor series expansion around $x=0$.  For example, 
$E_1(x)=1 + x$, $E_{n+1}(x)=1 + \int_0^x E_n(t)$, etc.
However, I wonder how this argument was developed informally before the proof.  For example, how was $E_1(x)$, etc. chosen?

A:

One way of solving a differential equation of the form
$$
\begin{align}
&\frac{dy}{dx}=F(y(x)),\\
&y(0)=a,
\end{align}
$$
is to rewrite it in integral form
$$
y(x)=a+\int_0^xF(y(u))\,du.
$$
Here, we are solving for functions $y\colon\mathbb{R}^+\to\mathbb{R}$ and $F\colon\mathbb{R}\to\mathbb{R}$ is given. The integral form can be solved iteratively. First choose any (continuous) initial guess $y_0\colon\mathbb{R}^+\to\mathbb{R}$ then iteratively define $y_{n+1}(x)$ in terms of $y_n$
$$
y_{n+1}(x)=a+\int_0^xF(y_n(u))\,du.
$$
This is method is known as Picard iteration, and is guaranteed to converge to the unique solution to the differential equation for a large class of functions $F$. For example, it always converges if $F$ is Lipschitz continuous.
The exponential function $y(x)=\exp(x)$ satisfies $\frac{dy}{dx}=y$ and $y(0)=1$. This differential equation can be solved by Picard iteration by taking $F(y)=y$ and using $y_0=0$ or $y_0=1$ gives the iteration described in the question.

