I was studying the finite difference scheme: $$u^{n+1}_j=\frac{u^{n}_{j+1}+u^{n}_{j-1}}{2}-\frac{c\delta t}{2 \delta x}(u^{n}_{j+1}+u^{n}_{j-1})+\delta t f^n_j.$$ for $n\in \mathbb{N}$ and $j \in \mathbb{Z}$. My question is that I tired to prove it's stability in $l^2$, by using Fourier transform, I had the following equality:
$$U^{n+1}_k=(cos(k.\delta x)-\frac{c.\delta t}{\delta x }.i.sin(k.\delta x))U^n_k+\delta tf^n_k$$ if $A=(cos(k.\delta x)-\frac{c.\delta t}{\delta x }.i.sin(k.\delta x))$ then we'll have: $$U^{n+1}_k=A.U^{n}_k+\delta tf^n_k$$ But I can't figure out how to prove:(for some $C>0$) $$||U^{n+1}_k||<C||U^{n}_k||$$