221 Completeness
A set where every sequence of elements that get closer to each other has a limit in the set that is used for finding the limit of a function.
definition [d] (Completeness) From Kreyszig Definition 1.4-3: the space \(X\) is said to be complete if every Cauchy sequence in \(X\) converges, that is, has a limit which is an element of \(X\).
where
- \(X = (X,d)\) is a metric space.
- a Cauchy sequence is as in Kreyszig Definition 1.4-3.
Note:
- Kreyszig Theorem 1.4-4: the real line and the complex plane are complete metric spaces.
221.1 Elementary Example
221.1.1 Simple
A metric space is complete when every Cauchy sequence converges in the space. The real line is complete.
\[ X = \mathbb{R},\quad d(x,y) = |x - y| \]
\[ x_{n} = \dfrac{1}{n} \text{ is Cauchy and } x_{n} \to 0 \in \mathbb{R} \]
where
- \(X\) is the metric space.
- completeness means every Cauchy sequence has a limit in \(X\).
221.1.2 General
The plane \(\mathbb{R}^{2}\) with the Euclidean metric is complete: every Cauchy sequence of points converges to a point of \(\mathbb{R}^{2}\).
\[ X = \mathbb{R}^{2},\quad d(x,y) = \sqrt{(x_{1}-y_{1})^{2} + (x_{2}-y_{2})^{2}} \]
where
- Kreyszig records that \(\mathbb{R}\) and \(\mathbb{C}\) are complete; the same holds for \(\mathbb{R}^{n}\).