In fact, every real number can be written as the limit of a sequence of rational numbers.

If f takes only positive values, it is the limit of a sequence of step functions.

Limit of a sequence and limit of a function: see below.

The limit of a sequence is said to be the fundamental notion on which the whole of analysis ultimately rests.

The limit of a sequence is unique.

It is this fact that allows us to write any real number as the limit of a sequence of decimals.

For the direct limit of a sequence of ultrapowers, see Ultraproduct.

And the final critical part of the proof is to remember that the limit of a sequence is unique.

The generalizations to other types of sequences are considered in the following section and the main page Limit of a sequence.

The limit of a sequence and the limit of a function are closely related.

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to".

In other words, the limit of a sequence of vector measures is a vector measure.

The standard word is also the limit of a sequence of words defined recursively as follows:

For sigma-additivity, one needs in addition that the concept of limit of a sequence be defined on that set.

This is impossible because z is the norm limit of a sequence of unit vectors.

Intuitively, the Kuratowski limit of a sequence of sets is where the sets "accumulate".

A formal proof of the lemma requires us to take the limit of a sequence of random variables, which is not done here.

It can be given a topology by defining the limit of a sequence of elements of D(U).

The pointwise limit of a sequence of continuous functions may be a discontinuous function, but only if the convergence is not uniform.

Let be a loop in which is uniform limit of a sequence of rectifiable loops in U with bounded length.

Because F is not totally ordered, this is not a limit of a sequence of partial sums, but rather of a net.

Therefore, both the limit of a filter and the limit of a net are conceptually the same as the limit of a sequence.

A finitely generated group is sofic if it is the limit of a sequence of sofic groups.

The concept of a limit of a sequence can be generalised to the idea of a limit of a topological net.

The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows: