Lax Equivalence Theorem

Suppose we have a numerical method

where is the spatial step, while superscripts denote time steps.
Then this converges in norm
if and only if it is:

1. Consistent
2. Stable

Proof ()

Fix a norm (any norm) and
Then since:

we get that the method is stable
if and only if

Suppose the method is also consistent.
Then, assuming , we have

for some constant
and local truncation errors
Since , we get , which shows convergence.