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:

$\ds \lim_{n \mathop \to \infty} {x_n - x_{m_n} } = 0$

Proof

From Subsequence of Cauchy Sequence in Normed Division Ring is 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:

$\ds \lim_{n \mathop \to \infty} {x_n - x_{m_n} } = 0$

$\blacksquare$