Subsequence is Equivalent to Cauchy Sequence

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

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

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

Then:
 * $\displaystyle \lim_{n \to \infty} {x_n - x_{m_n}} = 0$

Proof
From Subsequence of a Cauchy Sequence is a Cauchy Sequence:
 * $\sequence {x_{m_n}}$ is a Cauchy sequence

Let $\epsilon > 0$.

By definition of a Cauchy sequence:
 * $\exists N: \forall n,m > N: \norm {x_n - x_m } < \epsilon$

From Index of Subsequence not Less than its Index: $\forall n \in \N : m_n \ge n$

Thus:
 * $\exists N: \forall n > N: \norm {x_n - x_{m_n} } < \epsilon$

By definition of convergence:
 * $\displaystyle \lim_{n \to \infty} {x_n - x_{m_n}} = 0$