I am trying to understand the stability of the forward Euler method. I read there's a model problem to see the stability.
$$y'(t) = \lambda y(t) \qquad t \in (0, \infty)$$ $$y(0) = 1$$
then the book shows this:
$$u_0 = 1 \qquad \text{does this come from the initial condition?}$$
then
$$u_{n+1}=u_n(1+\lambda h) = (1+\lambda h)^{n+1}, \qquad n ≥ 0$$
I cannot understand how $(1+\lambda h)^{n+1}$ appeared
EDIT
then the analysis says that the only way the $$\lim_{n \rightarrow \infty} u_n = 0$$ is that $$|1+\lambda h| < 1$$ (why does it need to be less than 1?) Which gives (according to the book) $$h < \frac{2}{|\lambda|}$$
for the inequallity $|1+\lambda h| < 1$ the thing I get is $\frac{-2}{\lambda} < h$ but still don't know how they get the absolute value of $\lambda$.
how does the book arrive to this conclusion? I don't expect a very complex answer (which is appreciated) but a simple way to understand the intermediate steps not shown.