In general, the following question arises:

Does the Euler method converge as

The answer is given in the following theorem. Before we state the theorem, following definitions are in order:

Definition: The local truncation error (or local error) associated with a given difference method is that quantity which fails to satisfy the exact solution of the difference equation.

Definition : The round-off error associated with a given method is that quantity which must be added to a finite representation of a computed number in order to make it the exact representation of that number.

Definition : We roughly define a method as convergent for a problem if, as more grid (mesh) points are taken, the numerical solution converges to the true solution in the absence of round-off errors.

We shall make these definitions more precise when specific classes of methods are discussed later.