$$ 2. Let $\left(x_{n} ight) $ and $\left(y_{n} ight) $ be Cauchy sequences of real numbers. Define $\left(x_{n} ight)$ to be equivalent to $\left(y_{n} ight), written $\left(x_{n} ight) \sim\left(y_{n} ight)$, if \lim _{n ightarrow \infty}\left|x_{n}-y_{n} ight|=0 \text {. } $$ Show that this defines an equivalence relation on the set of all Cauchy sequences of real numbers, Furthermore, prove that if $\left(x_{n} ight) $ and $\left(y_{n} ight)$ are Cauchy sequences, then $$ || x_{n}-y_{n}|-| x_{m}-y_{m}|| \leq\left|x_{n}-x_{m} ight|+\left|y_{m}- y_{n} ight| Conclude that $\left(\left|x_{n}-y_{n} ight| ight)$ is a Cauchy sequence. CS. JG. 060 $$