The following equations are part of a proof showing how the global error $E_i$ can be related to the truncation error $T_i$ when solving initial value problems. $L$ is the Lipschitz constant.
If $|E_{i+1} \leq |E_i|(1+hL) + h(T_i)$
then it follow by induction that
$|E_i| \leq \frac{T}{L}[(1 + hL)^i - 1]$, $i = 0, 1, 2,...,N$
I can see that for the base case when $i = 0$, i.e. iteration $0$ where the global error and truncation error are zero, that we will have $0 = 0$ and the statements holds but I can't prove the inductive case.
I always have problems with induction, does anyone know how induction can be performed here?
