Let $\{x_n\}$ and $\{y_n\}$ sequences in a metric space $(X,d)$. If $\{x_n\}$ is Cauchy sequence and $\displaystyle\lim_{n\to\infty}d(x_n,y_n)=0$, show that $\{y_n\}$ is a Cauchy sequence.
My proof: Let $\epsilon >0$. Since $\{x_n\}$ is a Cauchy sequence, there exists $N_1\in\mathbb{Z}^+$ such that, if $m,n\geq N_1$ then $d(x_m,x_n)<\frac{\epsilon}{3}$. $\displaystyle\lim_{n\to\infty}d(x_n,y_n)=0$ implies that there exists $N_2\in\mathbb{Z}^+$ such that, if $n\geq N_2$, then $d(x_n,y_n)<\frac{\epsilon}{3}$. Let $N=\max\{N_1,N_2\}$ and for $n,m\geq N$ we have $$d(y_m,y_n)\leq d(y_n,x_n)+d(x_n,x_m)+d(x_m,y_m)<\epsilon$$then $\{y_n\}$ is a Cauchy sequence.
Is my proof correct? thanks!!!