If $(x_n)$ and $(y_n)$ are positive real sequences such that $(y_n)$ is bounded and $\lim\frac{(x_n)}{(y_n)} = 0$, prove that $\lim(x_n) = 0$
So since $(y_n)$ is bounded, there exists a real number $M$ such that for all n natural numbers, $(y_n) < |M|$
So by the limit definition :
$\lim|\frac{(x_n)}{(y_n)}-0| = \lim\frac{(x_n)}{(y_n)} < \epsilon$
So $|(x_n)| < \epsilon(y_n)$
Now I need to find a natural number $N$ such that for all $n > N$, $|(x_n)| < \epsilon$ is this correct which would prove this statement. However I'm having trouble picking something that will work here? Am I on the right track? I have to be very careful also about anything I use to quote the theorem since this is an introduction to real analysis class.
Thanks!