Definition in book:
A difference method is $0$-stable if there are constants $h_0$ and $K$ such that for any mesh functions $\vec{x}_h$ and $\vec{z}_h$ with $h\leq h_0,$ we have
$|\vec{x}_n-\vec{z}_n|\leq K\{|\vec{x}_0-\vec{z}_0|+\max_{1\leq j\leq N}|\mathcal{N}_h\vec{x}_h(t_j)-\mathcal{N}_h\vec{z}_h(t_j)|\},\, 1\leq n\leq N$
where $\mathcal{N}_h\vec{u}(t_n)\equiv \frac{\vec{u}(t_n)-\vec{u}(t_{n-1})}{h_n}-f(t_{n},\vec{u}(t_n))$ is the difference operator for the backward Euler method
My proof attempt:
Let $\vec{s}_n=\vec{x}_n-\vec{z}_n,$ and $\theta=\max_{1\leq j\leq n}|\mathcal{N}_h\vec{x}_h(t_j)-\mathcal{N}_h\vec{z}_h(t_j)|$
so by definition of backward euler
$\implies\forall n\,\,\, \theta\geq|\frac{\vec{s}_n-\vec{s}_{n-1}}{h_n}-(f(t_n,\vec{x}_n)-f(t_n,\vec{z}_n))|$
by triangle inequality
$\implies \theta\geq\frac{|\vec{s}_n|}{h_n}-|\frac{\vec{s}_{n-1}}{h_n}+(f(t_n,\vec{x}_n)-f(t_n,\vec{z}_n))|$
rearranging
$\implies |\vec{x}_n-\vec{z}_n|\leq |\vec{s}_{n-1}+h_n(f(t_n,\vec{x}_n)-f(t_n,\vec{z}_n))|+h_n\theta$
by Lipschitz condtion
$\implies |\vec{x}_n-\vec{z}_n|\leq |\vec{s}_{n-1}|+h_nL|\vec{s}_n|+h_n\theta$
Rearranging again and attempting to bound as much as I can until I get stuck, but I know I am in the right track since the Forward Euler method follows similarly to this proof regarding $0$-stability. I will add what they did just to show the similarities and hopefully one of you can show me what I can do
$\implies |\vec{x}_n-\vec{z}_n|\leq\frac{1}{(1-h_nL)}|\vec{s}_{n-1}|+\frac{h_n}{(1-h_nL)}\theta$
$\leq\frac{1}{(1-h_nL)}[\frac{1}{(1-h_{n-1}L)}|\vec{s}_{n-2}|+\frac{h_{n-1}}{(1-h_{n-1}L)}\theta]+\frac{h_n}{(1-h_nL)}\theta$
$.$ $.$ $.$
$\leq\frac{1}{(1-h_1L)}...\frac{1}{(1-h_{n-1}L)}\frac{1}{(1-h_nL)}|\vec{x}_0-\vec{z}_0|+\theta\sum_{j=1}^{n}h_j\frac{1}{(1-h_jL)}\frac{1}{(1-h_{j+1}L)}...\frac{1}{(1-h_nL)}$
Now, I will skip to the last portion for proof of $0$-stability for the Forward Euler method so you get an idea of what I'm trying to do. It's pretty straight forward, all I need is a less than or equal to expression that bounds $\frac{1}{(1-hL)}$
$|\vec{s}_n|\leq(1+h_nL)|\vec{s}_{n-1}|+h_n\theta\leq(1+h_nL)[(1+h_{n-1}L)|\vec{s}_{n-2}|+h_{n-1}\theta]+h_n\theta\leq$
...$\leq(1+h_1L)...(1+h_{n-1}L)(1+h_nL)|\vec{s}_0|+\theta\sum_{j=1}^{n}h_j(1+h_{j+1}L)...(1+h_nL)\leq e^{Lt_n}|\vec{s}_0|+\frac{1}{L}(e^{Lt_n}-1)\theta$
where the last inequality was obtained by noting that $1+hL\leq e^{Lh}$
I can add more detail into how they derived this if you want, just add comment below, but they go on to say that the stability bound is satisfied, with $K=\max\{e^{Lb},\frac{1}{L}(e^{Lb}-1)\}$
Please numerical analysis gods, I need your input so don't be shy