I'm looking for the motivation of the core idea to a proof of the global existence version of Picard-Lindelöf.
Global Picard-Lindelöf. Let $f:[a,b] \times \mathbb{R}^n \to \mathbb{R}^n$ be continuous and (globally) Lipschitz in $y$. Then there is a global solution $y:[a,b] \to \mathbb{R}^n$ for each $y_0 \in \mathbb{R}^n$ to the ODE $$ y' = f(x,y), \ y(a) = y_0.$$
The main idea of one proof is to use Banach's Fixed Point Theorem on a special Banach Space, namely some sort of $\mathcal{C}([a,b], \mathbb{R}^n)$ with a weighted supremum-norm $$ ||y|| := \sup_{x \in [a,b]} e^{-\beta x} |y(x)|$$ with Lipschitz constant $L$ and $\beta > L$.
So my question: Where did that norm come from? What's the motivation to that? Is it just some space that I will become more familiar with in my further studies? And even so - where did it come from then?