Convergent Sequence is Cauchy Sequence/Normed Division Ring/Proof 1

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

Every convergent sequence in $R$ is a Cauchy sequence.

Proof
Let $\sequence {x_n}$ be a sequence in $R$ that converges to the limit $l \in R$.

Let $\epsilon > 0$.

Then also $\dfrac \epsilon 2 > 0$.

Because $\sequence {x_n}$ converges to $l$, we have:
 * $\exists N: \forall n > N: \norm {x_n - l} < \dfrac \epsilon 2$

So if $m > N$ and $n > N$, then:

Thus $\sequence {x_n}$ is a Cauchy sequence.

Also see

 * Real Convergent Sequence is Cauchy Sequence
 * Complex Sequence is Cauchy iff Convergent