In other words, the weak and strong topologies share the same convergent sequences.

One can show that a convergent sequence has only one limit.

The space of convergent sequences c is a sequence space.

It therefore makes little or no sense to talk of a convergent sequence of germs.

For example, the matrices whose column sums are absolutely convergent sequences form a ring.

This interpretation can be compared with the notion of a convergent sequence in analysis, where each element is more specific than the preceding one.

Geometrically, the space consists of a series of convergent sequences.

In numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence.

That is, the operation of taking Cesàro means preserves convergent sequences and their limits.

In this context, directed subsets again provide a generalization of convergent sequences.

The limit of the convergent sequence (a) is normally denoted .

Convergent sequences also can be considered as real-valued continuous functions on a special topological space.

Furthermore, the space of convergent sequences c is the image of cs under T.

The only convergent sequences or nets in this topology are those that are eventually constant.

A tauberian theorem states, under some growth condition, that the domain of L is exactly the convergent sequences and no more.

Notice that every convergent sequence is a Cauchy sequence.

This is similar to the first convergent sequence above, except that now the ratio of two terms is not fixed at exactly 1/2:

Every convergent sequence in such a space is eventually constant, hence every set is sequentially open.

The exponential objects are equipped with the (convergent sequence)-open topology.

If is the least value for which such a convergent sequence exists, then it is called the border rank of .

In a first-countable space (such as a metric space), it is enough to consider only convergent sequences, instead of all nets.

However, in the cocountable topology all convergent sequences are eventually constant, so limits are unique.

Since every convergent sequence is bounded, c is a linear subspace of ℓ.

Every weakly convergent sequence in is strongly convergent.

In a convergent sequence, the absolute difference between the value of the current element and the limit, will decrease as the sequence progresses.