If A is a nonsingular matrix and a numerically stable algorithm is used to compute the solution

Question:

If A is a nonsingular matrix and a numerically stable algorithm is used to compute the solution to a system Ax = b, then the relative error in the computed solution will always be small.
In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true.