219 Cauchy Sequence
A sequence of elements whose mutual distances tend to zero that is used to test for a limit without first naming that limit.
Note: Also called fundamental sequence.
definition [d] (Cauchy Sequence) From Kreyszig Definition 1.4-3: a sequence \((x_{n})\) in a metric space \(X = (X,d)\) is said to be Cauchy if for every \(\epsilon > 0\) there is an \(N = N(\epsilon)\) such that
- \(d(x_{m}, x_{n}) < \epsilon\)
for every \(m, n > N\).
where
- \(X = (X,d)\) is a metric space.
- \((x_{n})\) is a sequence in \(X\).
- \(\epsilon > 0\) is a real number.
- \(N = N(\epsilon)\) is a natural number depending on \(\epsilon\).
- \(d\) is the metric on \(X\).
219.1 Elementary Example
219.1.1 Simple
A Cauchy sequence has terms that eventually lie close to each other. On the line, the sequence \(1/n\) is Cauchy.
\[ X = \mathbb{R},\quad d(x,y) = |x - y| \]
\[ x_{n} = \dfrac{1}{n},\quad n \in \{ 1,\ 2,\ 3,\ 4,\ 5 \} \]
\[ d(x_{4},x_{5}) = \left|\dfrac{1}{4} - \dfrac{1}{5}\right| = \dfrac{1}{20} \]
where
- \((x_{n})\) is the sequence.
- \(d\) is the metric on \(X\).
219.1.2 General
In \(\mathbb{R}^{3}\) with the Euclidean metric, \(x_{n} = (1/n, 1/n, 1/n)\) is Cauchy because all pairwise distances tend to zero.
\[ d(x_{m},x_{n}) = \sqrt{3}\,\left|\dfrac{1}{m} - \dfrac{1}{n}\right| \]
where
- \(x_{n} \in \mathbb{R}^{3}\).
- for every \(\epsilon > 0\) there is \(N\) such that \(m,n > N\) implies \(d(x_{m},x_{n}) < \epsilon\).