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:

$\sequence {\paren {x_{K + n} }^{-1} }_{n \mathop \in \N}$ 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 $C \in \R_{>0}: \forall n > K: C < \norm {x_n}$

or equivalently:

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

By Norm Axiom $\text N 1$: Positive Definiteness:

$\forall n > K : x_n \ne 0$

Let $\sequence {y_n}$ be the subsequence of $\sequence {x_n}$ defined as:

$y_n = x_{K + n}$

By Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence:

$\sequence {y_n}$ is a Cauchy sequence.

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


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 $\forall n, m > N$:

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

Hence:

\(\ds \norm { {y_n}^{-1} - {y_m}^{-1} }\) \(<\) \(\ds \dfrac {\norm {y_n} } C \norm { {y_n }^{-1} - {y_m}^{-1} } \dfrac {\norm {y_m} } C\) $(1)$ above
\(\ds \) \(=\) \(\ds \dfrac 1 {C^2} \paren {\norm {y_n} \norm { {y_n }^{-1} - {y_m}^{-1 } } \norm {y_m} }\)
\(\ds \) \(=\) \(\ds \dfrac 1 {C^2} \norm {y_n \paren { {y_n }^{-1 } - {y_m }^{-1} } y_m}\) Norm Axiom $\text N 2$: Multiplicativity
\(\ds \) \(=\) \(\ds \dfrac 1 {C^2} \norm {\paren {y_n {y_n }^{-1 } - y_n {y_m }^{-1} } y_m }\) Ring Axiom $\text D$: Distributivity of Product over Addition
\(\ds \) \(=\) \(\ds \dfrac 1 {C^2} \norm {y_n {y_n}^{-1} y_m - y_n {y_m}^{-1} y_m}\) Ring Axiom $\text D$: Distributivity of Product over Addition
\(\ds \) \(=\) \(\ds \dfrac 1 {C^2} \norm {y_m - y_n}\) Inverse Property of a Division Ring
\(\ds \) \(<\) \(\ds \dfrac 1 {C^2} \epsilon'\) $(2)$ above
\(\ds \) \(=\) \(\ds \dfrac 1 {C^2} \paren {\epsilon C^2}\) Definition of $\epsilon'$
\(\ds \) \(=\) \(\ds \epsilon\)

So:

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

$\blacksquare$


Sources

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