Is this limit correct? In $\mathbb R$, Suppose $\{x_n\}\to 0 $ and $\{y_n\}$ is bounded.
In order to prove $\lim_{n\to \infty}x_ny_n=0$, I proved:
$\lim_{n\to \infty}|x_ny_n|\leq M\lim_{n\to\infty}x=0$, and conclude the limit without absolute value exists, Is it correct?

Yeah the proof is correct if you say that $M$ is a bound of $y_n.$ 
For a more formal proof you can say: 
$$ - M |x_n|\leq  y_n |x_n| \leq M |x_n| $$
and with the sandwich theorem you have the convergence. 
