R^n, the set of n-tuples of real numbers, and l_2, the set of square-summable sequences, are complete.
Q, the set of all rational numbers, is not complete. For example, the sequence
3, 3.1, 3.14, 3.141, 3.1415, 3.14159...
where each term is a further approximation to pi, is Cauchy in Q but does not converge to a rational number.