= Solution
Let $e_n=y^n-y(t_n)$. Subtracting the exact one-step relation from the numerical method and applying the stated <Lipschitz continuity> gives
$$
\|e_{n+1}\|
\leq(1+hL)\|e_n\|+\|\tau_{n+1}\|.
$$
If $\|\tau_{n+1}\|\leq Ch^{p+1}$, iteration yields the <discrete Gronwall inequality>
$$
\|e_n\|
\leq(1+hL)^n\|e_0\|
+Ch^{p+1}\sum_{j=0}^{n-1}(1+hL)^j.
$$
For $nh\leq t^*$,
$$
(1+hL)^n\leq e^{nhL}\leq e^{t^*L}
$$
and
$$
h^{p+1}\sum_{j=0}^{n-1}(1+hL)^j
\leq\frac{e^{t^*L}-1}{L}h^p.
$$
Consequently
$$
\boxed{
\max_{0\leq n\leq\lfloor t^*/h\rfloor}
\|y^n-y(nh)\|
\leq e^{t^*L}\|e_0\|+O(h^p)}.
$$
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