Convergent sequence

Lets U = {un} a sequence of points in a normed space M. It is said that the sequence U converges to some u if

∀ ε >0, ∃ n0 >0 : if n > n0 => || un - u|| < ε

That is, for every number ε, however small we can find some n0 such as the points of sequence from this n0 tal que los puntos de la sucesión a partir de este n0are closer to the value ??ε.

Note: It is also said that a convergent sequence has a limit and its value is u.

The concept of convergence is closely related to the continuity and the topology of metric spaces, namely, every continuous function transforms convergent sequences into convergent sequences.