Product of Sequence Converges to Zero with Cauchy Sequence Converges to Zero

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

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

Let $\sequence {x_n}$ converge to $0$.

Let $\sequence {y_n}$ be a Cauchy sequence.

Then:
 * $\sequence{x_n y_n}$ and $\sequence{y_n x_n}$ converge to $0$.

Proof
By Cauchy Sequence is Bounded:
 * $\exists M \in \R_{\gt 0}: \forall n, \norm {x_n} \le M$

Given $\epsilon \gt 0$.

Since $\sequence {x_n}$ converges to $0$ then:
 * $\exists N \in \N: \forall n \gt N, \norm {x_n} \lt \dfrac \epsilon M$

Hence:

Similarly, $\norm {y_n x_n - 0} < \epsilon$

The result follows from convergence in normed division rings.