 Solution

For $0\le x\le 1$ and every real $z$,where $2|z|\le 1+z^2$ was used. The mean value theorem therefore proves Lipschitz continuity in $y$ with constant $1$.

Let $e_n=y(x_n)-y_n$. The explicit Euler method and the exact-solution identity from part (i) giveUsing the requested local truncation error bound $|{\tau }_n|\le h/2$, including at $n=0$, yieldsIteration and the finite geometric series giveSince $1+h\le e^h$ and $Nh=1$,Keeping the stronger bound $|{\tau }_n|\le h/4$ from part (i) improves the same proof to $(e-1)h/4$. The requested inequality follows already from the weaker bound.