Equivalence Relation on Cauchy Sequences

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Lemma

Let $\struct {X, d}$ be a metric space.

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

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

$\displaystyle \sequence {x_n} \sim \sequence {y_n} \iff \lim_{n \mathop \to \infty} d \paren {x_n, y_n} = 0$


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 $\sequence {x_n}, \sequence {y_n}, \sequence {z_n} \in \mathcal C \left[{X}\right]$ be arbitrary.


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

Therefore $\displaystyle \lim_{n \mathop \to \infty} d \paren {x_n, x_n} = 0$.

This shows that $\sequence {x_n} \sim \sequence {x_n}$.

Thus $\sim$ is reflexive.

$\Box$


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

Therefore:

$\displaystyle \lim_{n \mathop \to \infty} d \paren {x_n, y_n} = \lim_{n \mathop \to \infty} d \paren {y_n, x_n}$

So $\sequence {x_n} \sim \sequence {y_n}$ implies that $\sequence {y_n} \sim \sequence {x_n}$.


Thus $\sim$ is symmetric.

$\Box$


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

Therefore, by the sum rule for limits of sequences:

$\displaystyle \lim_{n \mathop \to \infty} d \paren {x_n, z_n} \le \lim_{n \mathop \to \infty} d \paren {x_n, y_n} + \lim_{n \mathop \to \infty} d \paren {y_n, z_n}$

Thus $\sequence {x_n} \sim \sequence {y_n}$ and $\sequence {y_n} \sim \sequence {z_n}$ together imply that $\sequence {x_n} \sim \sequence {z_n}$.


Thus $\sim$ is transitive.

$\Box$


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

$\blacksquare$