Theorem 1. (Existence theorem): Suppose that $f(x, y)$ is a continuous function in some region
$$R = \{(x, y) : |x − x_0| ≤ a, |y − y_0| ≤ b\}\qquad a, b > 0$$
Since $f$ is continuous in a closed and bounded domain, it is necessarily bounded in $R$, i.e., there exists $K > 0$ such that $|f(x, y)| \leq K$ $\forall (x, y) \in R$. Then the IVP $$ y'=f(x,y), \qquad y(x_0)=y$$ has at least one solution $y = y(x)$ defined in the interval $|x − x_0| \leq \alpha$ where $α = \min(a , b/K )$.
Why is the interval for finding solution $x=\min(a,b/K)$? Particularly why is there $b/k$ and not just $b$?