real analysis - If $K = lim_{i to infty}frac{x_i}{y_i}$ and $lim_{i to infty} x_i = 0$, then $lim_{i to infty} y_i =0$?

If $$K = \lim_{i \to \infty}\frac{x_i}{y_i}$$
with $0and $$\lim_{i \to \infty} x_i = 0$$
then will it be the case that
$$\lim_{i \to \infty} y_i =0$$
? I tried to prove this using different properties of limit, but so far I found no way. L'hopital's rule applies only when we can assume limits of both denominator and numerator exists or are infinite...

Edit: the main problem I observe in proving and disproving is eliminating/confirming possibility where the limit of $y_i$ as $i$ goes to infinity does not exist.

Answer

Since $\lim_{i \to \infty}\frac{x_i}{y_i} = K$ and $0 < K < \infty$ we have that $\lim_{i \to \infty}\frac{y_i}{x_i} = \frac{1}{K} = L$, where $0 < L < \infty$

Then we have:

$$\lim_{i \to \infty} y_i = \lim_{i \to \infty} \frac{y_i}{x_i} \cdot \lim_{i \to \infty} x_i = L \cdot 0 = 0$$