Combination Theorem for Sequences/Normed Division Ring/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 sequence in $R$.

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

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

Suppose $l \ne 0$.


Then:

$\exists k \in \N : \forall n \in \N : x_{k+n} \ne 0$.

and the subsequence $\sequence { x_{k+n}^{-1} }$ is well-defined and convergent with:

$\displaystyle \lim_{n \mathop \to \infty} x_{k+n}^{-1} = l^{-1}$.


Proof

Since $\sequence {x_n}$ converges to $l \ne 0$, by Sequence Converges to Within Half Limit then:

$\exists k \in \N: \forall n \in \N: \dfrac {\norm l} 2 \lt \norm {x_{k+n}}$.

By Axiom (N1) of norm (Positive definiteness) then:

$\forall n \in \N : x_{k+n} \ne 0$.


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

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

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

It also follows that:

$\forall n \in \N : y_n \ne 0$.

So $\sequence { {y_n}^{-1} }$ is well-defined and $\sequence { {y_n}^{-1} } = \sequence { x_{k+n}^{-1} }$.

Lemma

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


$\blacksquare$


Sources