4 - Complete Metric Spaces

Summary

Convergent sequences

The sequence of points {a_n} in a metric space M is said to converge to a point a of M if the distance ρ(a_n, a) tends to zero as n → ∞, that is, if for every positive value of ∈ there exists an integer n₀, depending on ∈, such that

whenever n ≥ n₀. The point a is called the limit of the sequence. We say the limit, because no sequence can converge to two limits; for if {a_n} converged to a and to b, we should have

as n → ∞, and so ρ(a, b) = 0 which is impossible since a and b are distinct.

In particular, if a_n = a for all but a finite number of values of n, the sequence {a_n} converges to a.

If {k_n} is a strictly increasing sequence of positive integers, so

that k_n ≥ n and k_n → ∞ as n → ∞, the sequence {ak_n} is called a subsequence of {a_n}; and if a_n → a.

We recall that a sequence is not a set. The elements in a sequence are ordered and are not necessarily distinct; the elements in a set are not ordered and are distinct. For example, we could define a sequence by a_2n = b, a_2n+1 = c where b ≠ c; the values taken by a_n form a set with only two members.