Let $f:[t_{0}-a,t_{0}+a] \times [y_{0}-b,y_{0}+b] \to \mathbb{R}$ be continuously differentiable. Consider the problem
$$\begin{align*} y' &= f(t,y),\\ y(t_{0})&=y_{0}. \end{align*}$$
Assume that the solution to the initial value problem exists in the interval $[t_{0}-a,t_{0}+a]$. Prove that Euler's method converges.
I have a theorem that will give convergence if I can establish that $$ |w_{i}-y(t_{i})| \le \frac{hM}{2L} \left(e^{L(t_{i}-a)}-1\right),$$ where $y_{i}$ is the true solution, $w_{i}$ is the approximation from Euler's method, $M$ is a bound on $|y''(t)|$ and $L$ is the Lipschitz constant for $f$ in the variable $y$.
I can obtain $M$ from the fact that $f$ has a continuous derivative on a compact set, but I am having trouble determining if $f$ is Lipschitz continuous in $y$.
How can I show that $f$ is Lipschitz in $y$?