Is the sequence $( d(x_{n},y_{n}))$ convergent if $X = \mathbb{R}$ with standard topology.

Question

Let $(X,d)$ be a metric space and $ ( x_{n} )$ , $ ( y_{n} )$ convergent sequences in $X$. Is the sequence $( d(x_{n},y_{n}))$ convergent if $X = \mathbb{R}$ with standard topology.

I am having trouble showing that it is convergent because it seems to me like the metric $d$ could be anything. Or does it have to be the absolute value on $\mathbb{R}$?

Answer 1

Actually, it always converges, no matter what is the metric that you are working with. In fact, if $\lim_{n\to\infty}x_n=x$ and $\lim_{n\to\infty}x_n=y$, then $\lim_{n\to\infty}d(x_n,y_n)=d(x,y)$.

Answer 2

Let $x, y$ be the respective limits. Then $$d(x_n, y_n) \leq d(x, x_n) +d(x, y) +d(y, y_n) $$ and $$d(x, y) \leq d(x, x_n) +d(x_n, y_n) +d(y, y_n) $$ so $$d(x_n, y_n) \geq d(x, y) - d(x, x_n) - d(y, y_n) $$ Since $d(x, x_n) \to 0$ and $d(y, y_n) \to 0$, these bounds imply that $d(x_n, y_n) \to d(x, y) $.