The sequences $\{x_n \},\{y_n \}$ are called concurrent if and only if $d(x_n,y_n) \to 0$ as $n \to \infty$. We denote this property as $\{x_n \} \approx \{y_n \}$. Prove if any two sequences converge to the same limit, they are concurrent.
I was thinking something like for any $\epsilon \gt 0, d(x,y) \lt \epsilon$, right? Since the two sequences converge to the same point, you know that for an $\epsilon \gt 0$ you will have $N_1$ and $N_2$ such that after those two indices, the sequences will be $\epsilon$-close to the limit point. Now take the max of the two, i.e. $\max(N_1,N_2)$. Then apply triangle inequality. Is this method correct ? Is there any other way to prove this question ?