Equivalence Relation on Cauchy Sequences

Lemma

Let $\left({X, d}\right)$ be a metric space.

Let $\mathcal C \left[{X}\right]$ denote the set of all Cauchy sequences in $X$.

Let a relation $\sim$ be defined on $\mathcal C \left[{X}\right]$ by:

$\displaystyle \left \langle {x_n} \right \rangle \sim \left \langle {y_n} \right \rangle \iff \lim_{n \to \infty} d \left({x_n, y_n}\right) = 0$

Then $\sim$ is an equivalence relation on $\mathcal C \left[{X}\right]$.

Proof

We must show that $\sim$ is

• reflexive,
• symmetric and
• transitive

on $\mathcal C \left[{X}\right]$.

To this end, let $\left \langle {x_n} \right \rangle, \left \langle {y_n} \right \rangle, \left \langle {z_n} \right \rangle \in \mathcal C \left[{X}\right]$ be arbitrary.

For each $n \in \N$ we have that $d \left({x_n, x_n}\right) = 0$ by metric space axiom M1.

Therefore $\displaystyle \lim_{n \to \infty} d \left({x_n, x_n}\right) = 0$.

This shows that $\left \langle {x_n} \right \rangle \sim \left \langle {x_n} \right \rangle$.

Thus $\sim$ is reflexive.

$\Box$

By metric space axiom M3, $d \left({x_n, y_n}\right) = d \left({y_n, x_n}\right)$ for each $n \in \N$.

Therefore:

$\displaystyle \lim_{n \to \infty} d \left({x_n, y_n}\right) = \lim_{n \to \infty} d \left({y_n, x_n}\right)$

So $\left \langle {x_n} \right \rangle \sim \left \langle {y_n} \right \rangle$ implies that $\left \langle {y_n} \right \rangle \sim \left \langle {x_n} \right \rangle$.

Thus $\sim$ is symmetric.

$\Box$

Finally, by metric space axiom M2, $d \left({x_n, z_n}\right) \le d \left({x_n, y_n}\right) + d \left({y_n, z_n}\right)$ for each $n \in \N$.

Therefore, by the sum rule for limits of sequences:

$\displaystyle \lim_{n \to \infty} d \left({x_n, z_n}\right) \leq\lim_{n \to \infty} d \left({x_n, y_n}\right) + \lim_{n \to \infty} d \left({y_n, z_n}\right)$

Thus $\left \langle {x_n} \right \rangle \sim \left \langle {y_n} \right \rangle$ and $\left \langle {y_n} \right \rangle \sim \left \langle {z_n} \right \rangle$ together imply that $\left \langle {x_n} \right \rangle \sim \left \langle {z_n} \right \rangle$.

Thus $\sim$ is transitive.

$\Box$

So $\sim$ is shown to be reflexive, symmetric and transitive, and therefore an equivalence relation on $\mathcal C \left[{X}\right]$.

$\blacksquare$