# Proving certain properties of limits of sequences using the definition of limits

I have been trying to prove the following properties for limits of sequences:

1. Let $$b \neq 0$$. If $$\lim x_n = a$$ and $$\lim \frac{x_n}{y_n} = b$$, then $$\lim y_n = \frac{a}{b}$$.
2. Let $$a \neq 0$$. If $$\lim x_n = a$$ and $$\lim x_ny_n = b$$, then $$\lim y_n = \frac{b}{a}$$.

I know that these are straightforward to prove using other elemental properties of limits of sequences, but I want to prove them using the definition and I have been not able to do it.

So far I have, for (1):

Let $$\epsilon > 0$$. We know that:

i)$$\exists n_1 \in \mathbb{N}: n >n_1 \implies |x_n - a| < \epsilon$$

ii)$$\exists n_2 \in \mathbb{N}: n >n_2 \implies |x_n/y_n - b| < \epsilon \implies |x_n - by_n| < |y_n|\epsilon$$

Now let $$n_0 = \max\{n_1, n_2\}$$. Then, for $$n > n_0$$:

\begin{aligned} |y_n - a/b| &= |1/b||by_n-a| \\ &= |1/b||by_n + x_n - x_n - a| \\ &\leq |1/b|(|x_n - by_n| + |x_n - a|) \\ &<|1/b|(|y_n| + 1)\epsilon \end{aligned}

which doesn't work, since it depends on the value $$|y_n|$$, so what I need to prove is that $$|y_n|$$ is bounded above by an $$M > 0$$. I know that both $$x_n$$ and $$x_n/y_n$$ are bounded since they are convergent, but I can't prove the last step that $$y_n$$ is bounded. Thanks in advance!

Because $$x_n/y_n \to b$$ and $$b$$ is nonzero, there is a positive integer $$W$$ so that whenever $$n \geq W$$ one has $$|x_n/y_n| \geq |b|/2$$.
(Please note, this is a step away from the "epsilon-N" definition of the assumed limit, but you get there by putting $$|b|/2$$ in for the "epsilon" in that definition and manipulating a few inequalities to see that $$|x_n/y_n - b| < |b|/2$$ implies $$|x_n/y_n - 0| \geq |b|/2$$. This is also clear from a picture of the number line. Anyway, you'll later want to toss $$W$$ into that $$\max$$ that you've got going.)
For all $$n \geq W$$ it follows that $$|y_n| \leq 2 |x_n|/|b|$$.
And if you know that $$x_n$$ that is bounded...