Completeness - Non-Completeness

• a metric space (𝑋,𝑑) in which every Cauchy sequence converges to an element that exists in 𝑋 is called complete
• all other metric spaces are called non-complete

Complete Metric Space

• is a type of mathematical space
• is a metric space (𝑋,𝑑) with completeness

Completeness Examples

Real Numbers

The real numbers are complete under the metric induced by the usual absolute value, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers. In this construction, each equivalence class of Cauchy sequences of rational numbers with a certain tail behavior—that is, each class of sequences that get arbitrarily close to one another— is a real number.

Discrete Metric Space

A metric space (𝑋,𝑑) where 𝑑 is defined as the discrete distance metric (where any two distinct points are at a distance 1 from each other). Any Cauchy sequence of elements of 𝑋 must be constant beyond some fixed point and converges to the eventually repeating term.

Closed Intervals

The closed interval 𝑋 = [0,2] is complete

Non-Completeness Examples

Rational Numbers

The rational numbers ℚ are not complete (for the usual distance):

There are sequences of rationals that converge to irrational numbers; these are Cauchy sequences having no limit in ℚ. In fact, if a real number 𝑥 is irrational, then the sequence (𝑥_𝑛), whose 𝑛^th term is the truncation to 𝑛 decimal places of the decimal expansion of 𝑥, gives a Cauchy sequence of rational numbers with irrational limit 𝑥. Irrational numbers certainly exist in ℝ, for example:

• The sequence defined by
  • $x_0=1,x_{n+1}=\frac{x_n+2/x_n}{2}$
• consists of rational numbers (1, 3/2, 17/12, …), which is clear from the definition; however it converges to the irrational square root of 2 which does not exist in ℚ.
• The sequence 𝑥_𝑛 = 𝐹_𝑛/𝐹_𝑛−1 of ratios of consecutive Fibonacci numbers which, if it converges at all, converges to a limit 𝜑 satisfying 𝜑²= 𝜑 + 1, and no rational number has this property. If one considers this as a sequence of real numbers, however, it converges to the real number 𝜑 = (1+√5)/2, the Golden ratio, which is irrational.
• The values of 𝑒𝑥𝑝(𝑥), 𝑠𝑖𝑛(𝑥), and 𝑐𝑜𝑠(𝑥), are known to be irrational for any rational value of 𝑥 ≠ 0, but each can be defined as the limit of a rational Cauchy sequence, using, for instance, the Maclaurin series.

Open Intervals

The open interval 𝑋 = (0,2) in the set of real numbers with an ordinary distance is not a complete space: there is a sequence 𝑥_𝑛 = 1/𝑛 in it, which is Cauchy (for arbitrarily small distance bound 𝑑 > 0 all terms 𝑥_𝑛 of 𝑛 > 1/𝑑 fit in the (0,𝑑) interval), however, does not converge in 𝑋 — its ‘limit’, number 0, does not belong to the space 𝑋.