Combination Theorem for Cauchy Sequences/Quotient Rule

Theorem
Let $\left({F, \left\Vert{\,\cdot\,}\right\Vert}\right)$ be a valued field with zero: $0$.

Let $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ be a Cauchy sequences in $F$.

Suppose $\left \langle {y_n} \right \rangle$ does not converge to $0$.

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

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

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

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

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

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

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

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

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

Then $\left \langle{ \dfrac {x'_n} {y'_n} }\right \rangle$ is well-defined and $\left \langle{ \dfrac {x'_n} {y'_n} }\right \rangle = \left \langle{ \dfrac {x_{K+n}} {y_{K+n}} }\right \rangle_{n \in \N}$.

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

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

Suppose $\left\Vert{x'_n}\right\Vert \le C_1$ for $C_1 \in \R$ and $n = 1, 2, 3, \ldots$.

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

Suppose $\left\Vert{y'_n}\right\Vert \le C_2$ for $C_2 \in \R$ and $n = 1, 2, 3, \ldots$.

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

Then the sequences $\left \langle {x'_n} \right \rangle$ and $\left \langle {y'_n} \right \rangle$ are both bounded by $C$.

Let $\epsilon > 0$ be given.

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

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

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

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

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

Hence:

Hence:
 * $\left \langle{ \dfrac {x'_n} {y'_n} }\right \rangle$ is a Cauchy sequence in $F$.