Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence

Theorem

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero: $0$.

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

Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$.

Then:

$\sequence {x_{n_r} }$ is a Cauchy sequence in $R$.

Proof

Let $\epsilon > 0$.

Since $\sequence {x_n}$ is a Cauchy sequence then:

$\exists N: \forall n,m > N: \norm {x_n - x_m } < \epsilon$

Now let $R = N$.

Then from Strictly Increasing Sequence of Natural Numbers‎:

$\forall r, s > R: n_r \ge r$ and $n_s \ge s$

Thus $n_r, n_s > N$ and so:

$\norm {x_{n_r} - x_{n_s} } < \epsilon$

The result follows.

$\blacksquare$