Combination Theorem for Cauchy Sequences/Quotient Rule

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

That is, $\struct {R, \norm {\, \cdot \,} }$ is a valued field.

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

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

Then:
 * $\exists K \in \N : \forall n > K : y_n \ne 0$.

and the sequence
 * $\sequence { \dfrac {x_{K+n}} {y_{K+n}} }_{n \in \N} $ is well-defined and a Cauchy sequence.

Proof
Since $\sequence {y_n}$ does not converge to $0$, by Cauchy Sequence Is Eventually Bounded Away From Zero then:
 * $\exists K \in \N$ and $c \in \R_{\gt 0}: \forall n \gt K: c \lt \norm {y_n}$.

or equivalently:
 * $\exists K \in \N$ and $c \in \R_{\gt 0}: \forall n \gt K: 0 \lt \dfrac 1 {\norm {y_n}} \lt \dfrac 1 c$.

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

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

By Subsequence of a Cauchy Sequence is a Cauchy Sequence then $\sequence {x'_n} $ is a Cauchy sequence.

Let $\sequence {y'_n}$ be the subsequence of $\sequence {y_n}$ where $y'_n = y_{K+n}$.

By Subsequence of a Cauchy Sequence is a Cauchy Sequence then $\sequence {y'_n}$ is a Cauchy sequence.

Then $\sequence { \dfrac {x'_n} {y'_n} } $ is well-defined and $\sequence { \dfrac {x'_n} {y'_n} } = \sequence { \dfrac {x_{K+n}} {y_{K+n}} }_{n \in \N}$.

We now show that $\sequence { \dfrac {x'_n} {y'_n} }$ is a Cauchy sequence.

Because $\sequence {x'_n}$ is a Cauchy sequence, it is bounded by Cauchy Sequence is Bounded.

Suppose $\norm {x'_n} \le C_1$ for $C_1 \in \R_{\gt 0}$ and $n = 1, 2, 3, \ldots$.

Because $\sequence {y'_n}$ is a is a Cauchy sequence, it is bounded by Cauchy Sequence is Bounded.

Suppose $\norm {y'_n} \le C_2$ for $C_2 \in \R_{\gt 0}$ and $n = 1, 2, 3, \ldots$.

Let $C = \max \set {C_1, C_2}$.

Then the sequences $\sequence {x'_n}$ and $\sequence {y'_n}$ are both bounded by $C$.

Let $\epsilon > 0$ be given.

Let $\epsilon' = \dfrac {\epsilon c^2} {2C}$, then $ \epsilon' > 0$.

Since $\sequence {x'_n}$ is a Cauchy sequence, we can find $N_1$ such that:
 * $\forall n, m > N_1: \norm {x'_n - x'_m} < \epsilon'$

Similarly, $\sequence {y'_n}$ is a Cauchy sequence, we can find $N_2$ such that:
 * $\forall n, m > N_2: \norm {y'_n - y'_m} < \epsilon'$

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

Thus $\forall n, m > N$:
 * $ 0 \lt \dfrac 1 {\norm {y'_n}},\dfrac 1 {\norm {y'_m}} \lt \dfrac 1 c$.
 * $ \norm {x'_n - x'_m} < \epsilon'$.
 * $ \norm {y'_n - y'_m} < \epsilon'$.

Hence:

Hence:
 * $\norm { \dfrac {x'_n} {y'_n} }$ is a Cauchy sequence in $F$.