Is the mapping $ d : X\times X \mapsto \mathbb {R} $ continuous? [duplicate]

Where $ (X, d) $ is a metric space. I want to prove it using sequential criteria. How do I tackle it?

Solution 1:

The product topology on $X\times X$ is metrizable. A metric is given by $D:(X\times X)\times(X\times X)\to\mathbb R$, defined by $$D((x_1,y_1),(x_2,y_2))=d(x_1,x_2)+d(y_1,y_2).$$

Now, let $(x_n,y_n)_n$ be an arbitrary convergent sequence in $X\times X$, with limit $(x,y)$. Consider the difference $d(x_n,y_n)-d(x,y)$. Using the triangle inequality multiple times gives us the following sequence of inequalities:

$$\begin{align}|d(x_n,y_n)-d(x,y)|&=|d(x_n,y_n)-d(x_n,y)+d(x_n,y)-d(x,y)|\\&\leq |d(x_n,y_n)-d(x_n,y)|+|d(x_n,y)-d(x,y)|\\&\leq d(y_n,y)+d(x_n,x)\\&=D((x_n,y_n),(x,y))\end{align}$$

Therefore $|d(x_n,y_n)-d(x,y)|$ goes to zero as $n$ grows, so $d$ is continuous. The same argument shows that the mapping $d$ is $1$-Lipschitz for this choice of product metric.