Equivalent Cauchy Sequences have Equal Limits of Norm Sequences

Theorem
Let $\struct {R, \norm { \, \cdot \, } }$ be a normed division ring.

Let $\sequence {x_n}$ and $\sequence {x_n}$ be Cauchy sequences in $R$.

Let $\displaystyle \lim_{n \to \infty} {x_n - y_n} = 0$

Then:
 * $\displaystyle \lim_{n \to \infty} \norm {x_n} = \lim_{n \to \infty} \norm {y_n}$

Proof
By Norm Sequence of Cauchy Sequence has Limit then $\displaystyle \lim_{n \to \infty} \norm {x_n}$ and $\displaystyle \lim_{n \to \infty} \norm {y_n}$ exist.

Let $l = \displaystyle \lim_{n \to \infty} \norm {x_n}$ and $m = \displaystyle \lim_{n \to \infty} \norm {y_n}$ then:

By reverse triangle inequality, for $n \in \N$ then:

By the Squeeze Theorem then:
 * $\displaystyle \lim_{n \to \infty} \paren { \big\lvert \norm{x_n } - \norm{y_n } \big\rvert } = 0$

So $\lvert l - m \rvert = 0$ and therefore $l = m$