I'd like to show that for the BTCS for the heat equation: $\frac{{v^{n+1}_{m}}-{v^{n}_{m}}}{k}=b\frac{{v^{n+1}_{m+1}}-2v^{n+1}_{m}+{v^{n+1}_{m-1}}}{h^2}$ the maximum principle $$sup_m||v^{n+1}_{m}|| \leq sup_m||v^{n}_{m}||$$ is satisfied iff $2b\mu \leq 1$.
I have rewritten and collected terms so that the method can be written: $$-b\mu v^{n+1}_{m+1}+(1+2b\mu) v^{n+1}_{m}-b\mu v^{n+1}_{m-1}= v^{n}_{m}$$ where $\mu=\frac{k}{h^2}$
Having trouble with formulating an argument though. Any thoughts are appreciated.