Combination Theorem for Cauchy Sequences/Inverse Rule

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Theorem

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

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

Suppose $\sequence {x_n}$ does not converge to $0$.

Then:

$\exists K \in \N : \forall n > K : x_n \ne 0$

and the sequence

$\sequence { \paren {x_{K+n}}^{-1} }_{n \in \N}$ is well-defined and a Cauchy sequence.


Proof

Since $\sequence {x_n}$ does not converge to $0$, by Cauchy Sequence Is Eventually Bounded Away From Non-Limit then:

$\exists K \in \N$ and $C \in \R_{\gt 0}: \forall n \gt K: C \lt \norm {x_n}$

or equivalently:

$\exists K \in \N$ and $C \in \R_{\gt 0}: \forall n \gt K: 1 \lt \dfrac {\norm {x_n}} C$

By Axiom (N1) of norm (Positive definiteness) $\forall n > K : x_n \ne 0$.

Let $\sequence {y_n}$ be the subsequence of $\sequence {x_n}$ where $y_n = x_{K+n}$.

By Subsequence of a Cauchy Sequence is a Cauchy Sequence then $\sequence {y_n} $ is a Cauchy sequence.

So $\sequence { {y_n}^{-1} } $ is well-defined and $\sequence { {y_n}^{-1} } = \sequence { \paren {x_{K+n}}^{-1} }_{n \in \N}$.


Let $\epsilon > 0$ be given.

Let $\epsilon' = \epsilon C^2$, then $ \epsilon' > 0$.

Similarly, $\sequence {y_n}$ is a Cauchy sequence, we can find $N$ such that:

$\forall n, m > N_2: \norm {y_n - y_m} < \epsilon'$

Thus $\forall n, m > N$:

$(1): \quad 1 \lt \dfrac {\norm {y_n}} C, \dfrac {\norm {y_m} } C$
$(2): \quad \norm {y_n - y_m} < \epsilon'$

Hence:

\(\displaystyle \norm { {y_n }^{-1 } - {y_m }^{-1 } }\) \(<\) \(\displaystyle \dfrac { \norm {y_n } } C \norm { {y_n }^{-1 } - {y_m }^{-1 } } \dfrac { \norm {y_m } } C\) $(1)$ above
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {C^2 } \paren { \norm {y_n } \norm { {y_n }^{-1 } - {y_m }^{-1 } } \norm {y_m } }\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {C^2 } \norm { y_n \paren { {y_n }^{-1 } - {y_m }^{-1 } } y_m }\) Axiom (N2) of norm: Multiplicativity
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {C^2 } \norm { \paren { y_n {y_n }^{-1 } - y_n {y_m }^{-1 } } y_m }\) Axion (D) of a ring: Distributivity
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {C^2 } \norm { y_n {y_n }^{-1 } y_m - y_n {y_m }^{-1 } y_m }\) Axion (D) of a ring: Distributivity
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {C^2} \norm {{y_m} - {y_n}}\) Inverse property of a division ring
\(\displaystyle \) \(\lt\) \(\displaystyle \dfrac 1 {C^2} \epsilon'\) $(2)$ above
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {C^2} \paren {\epsilon C^2 }\) Definition of $\epsilon'$
\(\displaystyle \) \(=\) \(\displaystyle \epsilon\)

So:

$\sequence { { {y_n}^{-1} } }$ is a Cauchy sequence in $R$.


$\blacksquare$


Sources