Cauchy sequence - Cauchy sequence in a metric space

In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. To be more precise, by dropping a finite number of elements from the start of the sequence we can make the distance between any two remaining elements arbitrarily small. Cauchy sequences require the notion of distance so they can only be defined in a metric space. Generalizations to more abstract uniform spa ...

Including:

• Cauchy sequence - Cauchy sequence in a metric space
• Cauchy sequence - Formal definition
• Cauchy sequence - Completeness
• Cauchy sequence - Other properties
• Cauchy sequence - Cauchy sequences in topological vector spaces
• Cauchy sequence - Cauchy sequences in groups

Read more here: » Cauchy sequence: Encyclopedia - Cauchy sequence

Cauchy sequence - Formal definition. Formally, a Cauchy sequence is a sequence in a metric space (M, d) such that for every positive real number r > 0, there is an integer N such that for all integers m,n > N, the distance d(xm,xn) is less than r. Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ough ...

See also:

Cauchy sequence, Cauchy sequence - Cauchy sequence in a metric space, Cauchy sequence - Formal definition, Cauchy sequence - Completeness, Cauchy sequence - Other properties, Cauchy sequence - Cauchy sequences in topological vector spaces, Cauchy sequence - Cauchy sequences in groups

Read more here: » Cauchy sequence: Encyclopedia II - Cauchy sequence - Cauchy sequence in a metric space