Completeness (real analysis)

In real analysis, a metric space $$X$$ is said to be complete if every Cauchy sequence has a limit that exists in $$X$$. An informal way to describe completeness is to consider a metric space that have no "missing points". For example, any sequence of real numbers will converge to some limit that is also a real number. So the space of real numbers is said to be complete. On the other hand, the field of rationals is not complete because one can construct a Cauchy sequence of rationals that "converge" to $$\sqrt 2$$ but $$\sqrt 2$$ is not an element inside the field of rationals. So in a sense, the field of rationals is "missing" the point $$\sqrt 2$$.

Definition
To give meaning to the definition, one should revisit what it means to have a Cauchy sequence. A sequence $$\{x_n\}$$ said to be Cauchy in the metric space $$(X, d)$$ if, for every $$\varepsilon > 0$$, there exists a positive constant $$N$$ such that, for any $$n, m > N$$, $$d(x_n, x_m) < \varepsilon$$. If every Cauchy sequence converges to a limit in $$X$$, then we say that $$(X, d)$$ is complete. Alternatively, every Cauchy sequence converges in $$X$$.

Examples

 * $$(\mathbb R, | \cdot |)$$ is complete.
 * $$(\mathbb Q, | \cdot |)$$ is not complete; a Cauchy sequence of rational numbers converges to $$\sqrt 2$$ but $$\sqrt 2$$ does not belong in $$\mathbb Q$$.
 * $$(\mathbb Z, | \cdot |)$$ is complete; any Cauchy sequence in $$\mathbb Z$$ converges to a constant sequence since $$|x_n - x_m| < 1$$ implies that $$x_n = x_m$$.
 * Let $$X = (0, 1)$$ with the usual absolute value metric. Then $$(X, d)$$ is not complete; consider the sequence $$x_{n} = \frac{1}{n}$$. It's not too difficult to show that it is a Cauchy sequence; however, it "converges" to 0 which does not belong in $$X$$.

Completion of metric spaces
While not every metric space may be complete, we can instead look at metric spaces as a dense subset of a complete metric space instead. The next theorem is an important classification of metric spaces.

The completion of a metric space is unique (up to isomorphism). In other words, we can begin to talk about the completion of a metric space.