The Picard–Lindelöf existence-uniqueness theorem

Rewriting the initial value problem as an integral equation, a continuous function $y(x)$ solves $y'=f(x,y),\ y(x_0)=y_0$ if and only if it solves $y(x) = y_0 + \int_{x_0}^x f(t,y(t))\,dt$; this equivalence follows from the fundamental theorem of calculus.

Define the Picard iteration operator $T$ on continuous functions by $T[y](x) = y_0 + \int_{x_0}^x f(t,y(t))\,dt$, restricted to a small interval $I=[x_0-h,x_0+h]$ chosen so that $f$ stays in the rectangle where the hypotheses hold.

For two continuous functions $y_1,y_2$ on $I$, the Lipschitz condition gives $|T[y_1](x)-T[y_2](x)| \le \int_{x_0}^x |f(t,y_1(t))-f(t,y_2(t))|\,dt \le L\,h\,\sup_{t\in I}|y_1(t)-y_2(t)|$. Choosing $h$ small enough that $Lh<1$ makes $T$ a contraction mapping in the supremum norm.

By the Banach fixed-point theorem, a contraction on a complete metric space (the continuous functions on $I$ with the supremum norm) has exactly one fixed point $y$, satisfying $y=T[y]$. This fixed point is precisely the unique solution of the original initial value problem on $I$.