TO prove that consistency is necessary and sufficient for convergence of
Runge--Kutta methods, we follow Henrici
henrici.
Let 
satisfy a Lipschitz
condition as in
Eq.(3) so that 
with 
has a unique
solution 
.
Using the mean value theorem:
Now define 
and 
.
Let 
then
We can write the part in parentheses as
Let
and
If 
is continuous and satisfies a Lipschitz condition, then
Thus
and so, since 1+hL and 
are both positive,
Now 
so 
. Thus
We now look at the fixed-station limit as in Eq.(28):
Since 
is continuous,
For the same reason
Thus we obtain
which shows that consistency is sufficient for convergence since if
then 
. To establish its necessity, let us assume
that the method is convergent but that 
at some point
. There exists an 
such that 
, the solution of the initial
value problem, passes through 
. 
as defined above then converges
in the limit to 
, and also, as we proved above, to 
. If
, then immediately there is a contradiction. Otherwise, if
then 
where 
and 
, but
which is again a contradiction. Thus we have proved
that consistency is necessary and sufficient for convergence, and it follows
from Eq.(30) that all Runge--Kutta methods are convergent.