Cauchy Sequences
The sequence members of a Cauchy Sequence eventually become arbitrarily close to each other.
Discover Bridges
Study Cauchy Sequences
Definition 1 (Cauchy sequences). A sequence \((a_n)_{n\in\mathbb{N}}\) in \(\mathbb{K}\) is called Cauchy sequence if for all \(\varepsilon>0\), there exists some \(N\) such that for all \(n,m\geq N\) holds \[|a_n-a_m|<\varepsilon.\]
Remark 2. By the expression “\(n,m\geq N\)”, we mean that both \(n\) and \(m\) are greater or equal than \(N\), i.e., \(n\geq N\) and \(m\geq N\).
Now we show that convergent sequences are indeed Cauchy sequences.
Theorem 3. Let \((a_n)_{n\in\mathbb{N}}\) be a convergent sequence. Then \((a_n)_{n\in\mathbb{N}}\) is a Cauchy sequence.
Proof: Let \(a=\lim_{n o\infty}a_{n}\) and \(\varepsilon>0\). Then there exists some \(N\) such that for all \(k\geq N\) holds \(|a-a_k|<\frac{\varepsilon}2\). Hence, for all \(m,n\geq N\) holds \[|a_n-a_m|=|(a_n-a)+(a-a_m)|\leq|a_n-a|+|a-a_m|< \frac{\varepsilon}2+\frac{\varepsilon}2=\varepsilon.\] \(\Box\)
We know that convergent sequences are bounded. The following theorem shows that this is also the case for Cauchy sequences.
Theorem 4 (Cauchy sequences are bounded). Let \((a_n)_{n\in\mathbb{N}}\) be a Cauchy sequence. Then \((a_n)_{n\in\mathbb{N}}\) is bounded.
Proof: Take \(\varepsilon=1\). Then there exists some
\(N\) such that for all \(n,m\geq N\) holds \(|a_n-a_m|<1\). Thus, for all \(n\geq N\) holds \[|a_n|=|a_n-a_N+a_N|\leq
|a_n-a_N|+|a_N|<1+|a_N|.\] Now choose \[c=\max\{|a_1|,|a_2|,\ldots,|a_{N-1}|,|a_N|+1\}\]
and consider some arbitrary sequence element \(a_k\).
If \(k<N\), we have that \(|a_k|\leq \max\{|a_1|,|a_2|,\ldots,|a_{N-1}|\}\leq
c\).
If \(k\geq N\), we have, by the above
calculations, that \(|a_k|<|a_N|+1\leq
c\).
Altogether, this implies that \(|a_k|\leq
c\) for all \(k\in\mathbb{N}\),
so \((a_n)_{n\in\mathbb{N}}\) is
bounded by \(c\).\(\Box\)
Courtesy of Marcus Waurick. Well-defined & Wonderful podcast, marcus-waurick.de.
Discuss your questions by typing below.
Solve the Exercise
The data for the interactive network on this webpage was generated with pntfx Copyright Fabian Gabel and Julian Großmann. pntfx is licensed under the MIT license. Visualization of the network uses the open-source graph theory library Cytoscape.js licensed under the MIT license.