Let $(X,d)$ is metric space. Let $d(y_n,z_n)\to r$. And let $(y_{m_k})$ and $(z_{m_k})$ be subsequences of $(y_n)$ and $(z_n)$ respectively, with $(y_{m_k})\to y\in X$ and $(z_{m_k})\to z\in X$. (Yes, the indices of subsequences are same.)
How to show that $$d(y,z)=d(\lim\limits_{k\to\infty} y_{m_k},\lim\limits_{k\to\infty} z_{m_k})=\lim\limits_{k\to\infty}d(y_{m_k},z_{m_k})=r?$$
I am having trouble deriving second of equality above (obviously).
I went line this: $d(\lim\limits_{k\to\infty} y_{m_k},z)=\lim\limits_{k\to\infty}d(y_{m_k},z)$ by continuity of $x\to d(x,a)$ for fixed $a$. But But I can't take out lim for both the $x$ and $y$ simultaneously.
