It's usually not really defined as the limit of a series in analysis texts, but as the simultaneous greatest lower bound and least upper bound of upper and lowe

It's usually not really defined as the limit of a series in analysis texts, but as the simultaneous greatest lower bound and least upper bound of upper and lower sums (inscribed and circumscribed rects area), respectively. THat def is a lot easier to deal with in proofs and just as intuitive to many.