Real numbers/Sequence/Limit and convergence/Definition

Convergent sequence

Let denote a real sequence, and let . We say that the sequence converges to , if the following property holds.

For every positive , , there exists some , such that for all , the estimate


If this condition is fulfilled, then is called the limit of the sequence. For this we write

If the sequence converges to a limit, we just say that the sequence converges, otherwise, that the sequence diverges.