# 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

holds.

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*.