# numerical methods – Maximum norm stability for implicit Black-Scholes equation

I am trying to prove maximum norm stability for the implicit Black-Scholes equation with a right-sided difference approximation given by

begin{align*} U_j^{(n+1)}&=-left(frac{sigma^2j^2Delta t}2right)U_{j-1}^{(n)}+left(1+r(1+j)Delta t+sigma^2j^2Delta tright)U_j^{(n)}-left(rjDelta tfrac{sigma^2j^2Delta t}2right)U_{j+1}^{(n)}\ &=a_jU_{j-1}^{(n)}+b_jU_j^{(n)}+c_jU_{j+1}^{(n)}. end{align*}

I am asked to prove that $$(1+rDelta t)max_j|U_j^{(n)}|leqmax|U_j^{(n+1)}|$$. All I know is that for backward, implicit equations like these I must satisfy $$a_j,,c_jleq0$$ and $$a_j+b_j+c_jgeq1$$, which is clearly satisfied here, but I don’t know how to justify the requested inequality. I know that the coefficient of the LHS is $$a_j+b_j+c_j$$, but given the negativity of two of the coefficients I am unclear of how to argue correctly.