Proof: To get an a priori bound, let us write
where  is called the local truncation error. It is the amount by which the solution fails to satisfy the difference method. Subtracting (2.2) from (2.1), we get
Let us write
Therefore,
This is a difference equation for  . The error  is known, so it can be solved if we know  and . We have a bound of the Lipschitz constant  for  . Suppose we also have  . Then we have
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