Cauchy Sequence in Positive Integers under Scaled Euclidean Metric

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Theorem

Let $\Z_{>0}$ be the set of (strictly) positive integers.

Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the scaled Euclidean metric on $\Z_{>0}$ defined as:

$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$


The sequence $\sequence {x_n}$ in $\Z_{>0}$ defined as:

$\forall n \in \N: x_n = n$

is a Cauchy sequence in $\struct {\Z_{>0}, \delta}$.


Proof

For a general $x_m, x_n \in \sequence {x_n}$ as defined:

\(\ds \map \delta {x, y}\) \(=\) \(\ds \frac {\size {x_m - x_n} } {x_m x_n}\) Definition of $\delta$
\(\ds \) \(=\) \(\ds \size {\frac 1 {x_m} - \frac 1 {x_n} }\) algebra
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds \size {\frac 1 m - \dfrac 1 n}\) Definition of $\sequence {x_n}$


Let $\epsilon \in \R_{>0}$.

Then by the Axiom of Archimedes:

$\exists N \in \N: N > \dfrac 1 \epsilon$

from which it follows that:

$\epsilon > \dfrac 1 N$


Thus:

\(\ds \forall m, n \in \N: \, \) \(\ds m, n\) \(>\) \(\ds N\)
\(\ds \leadsto \ \ \) \(\ds \map \delta {x_m, x_n}\) \(=\) \(\ds \size {\frac 1 m - \frac 1 n}\) from $(1)$ above
\(\ds \) \(<\) \(\ds \max \set {\frac 1 m, \frac 1 n}\)
\(\ds \) \(<\) \(\ds \frac 1 N\)
\(\ds \) \(<\) \(\ds \epsilon\)


Therefore $\sequence {x_n}$ is a Cauchy sequence in $\struct {\Z_{>0}, \delta}$.

$\blacksquare$


Sources

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