= Solution
For $0<\eta\leq h$, work on the closed subset
$$
C_\eta=\{y\in C([a-\eta,a+\eta]): |y(x)-b|\leq k\},\qquad d(y,z)=\|y-z\|_\infty.
$$
It is a nonempty <complete metric space> because it is closed in the <Banach space> of continuous functions with the <supremum norm>. Define the <Picard iteration> operator
$$
(Ty)(x)=b+\int_a^x f(s,y(s))\,ds.
$$
Choose $\eta$ so that \b[$M\eta\leq k$ and $L\eta<1$], omitting a bound whose constant is zero. Such a positive $\eta$ exists. The first inequality makes $T$ a self-map, since $|(Ty)(x)-b|\leq M|x-a|\leq k$. The <Lipschitz condition> gives
$$
\|Ty-Tz\|_\infty\leq L\eta\|y-z\|_\infty,
$$
so the <contraction mapping theorem> yields a unique fixed point. The <fundamental theorem of calculus> makes it a differentiable solution of $y'=f(x,y)$ with $y(a)=b$; conversely every solution staying in the rectangle satisfies the integral equation and is a fixed point. This is the local <Picard-Lindelöf theorem> construction. The interval-restricted space $C_\eta$ avoids treating functions defined on a shorter interval as elements of $C_h$.
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