Combination Theorem for Sequences/Normed Division Ring/Inverse Rule/Lemma

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Theorem

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

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

Let $\sequence {x_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limit:

$\displaystyle \lim_{n \mathop \to \infty} x_n = l$

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


Suppose $l \ne 0$, and for all $n: y_n \ne 0$.


Then:

$\displaystyle \lim_{n \mathop \to \infty} y_n^{-1} = l^{-1}$


Proof

By Limit of Subsequence equals Limit of Sequence then $\sequence {y_n}$ is convergent with:

$\displaystyle \lim_{n \mathop \to \infty} y_n = l$


Let $\epsilon > 0$ be given.

Let $\epsilon' = \dfrac {\epsilon {\norm l}^2 } {2}$.

Then:

$ \epsilon' > 0$


As $\sequence {y_n} \to l$, as $n \to \infty$, we can find $N_1$ such that:

$\forall n > N_1: \norm {y_n - l} < \epsilon'$


As $\sequence {y_n}$ converges to $l \ne 0$, by Sequence Converges to Within Half Limit:

$\exists N_2 \in \N: \forall n > \N_2: \dfrac {\norm l} 2 < \norm {y_n}$

or equivalently:

$\exists N_2 \in \N: \forall n > \N_2: 1 < \dfrac {2 \norm {y_n} } {\norm l}$


Let $N = \max \set {N_1, N_2}$.


Then $\forall n > N$:

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


Hence:

\(\displaystyle \norm { {y_n }^{-1} - l^{-1} }\) \(<\) \(\displaystyle \dfrac {2 \norm {y_n} } {\norm l} \norm { {y_n}^{-1} - l^{-1} }\) from $(1)$
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 2 {\norm l^2} \paren {\norm { {y_n } } \norm { {y_n}^{-1} - l^{-1} } \norm l}\) multiplying and dividing by $\norm l$
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 2 {\norm l^2} \norm {y_n \paren { {y_n}^{-1} - l^{-1} } l}\) Axiom (N2) of Norm: Multiplicativity
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 2 {\norm l^2} \norm {y_n y_n^{-1} l - y_n l^{-1} l}\) Axiom (D) of Ring: Distributivity
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 2 {\norm l^2} \norm {l - y_n}\) Inverse Property of Division Ring
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(<\) \(\displaystyle \dfrac 2 {\norm l^2} \epsilon'\) from $(2)$
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {2} {\norm l^2} \paren { \dfrac {\epsilon \norm l^2 } 2}\) Definition of $\epsilon'$
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(=\) \(\displaystyle \epsilon\) cancelling terms


Hence:

$\sequence { {y_n}^{-1} }$ is convergent with $\displaystyle \lim_{n \mathop \to \infty} {y_n}^{-1} = l^{-1}$.

$\blacksquare$