Initial value problems: existence and uniqueness

Picard–Lindelöf: if $f$ is Lipschitz in $y$, then $y' = f(t, y)$, $y(t_0) = y_0$ has exactly one solution near $t_0$. Determinism, mathematically: the present determines the future — and a simulation has a single right answer to approximate.

Rewrite as $y(t) = y_0 + \int_{t_0}^t f(s, y(s))\,\dd s$ and show the right-hand side is a contraction (Banach fixed point): Picard iteration converges.