Suppose $\displaystyle (\mathbb{X},d)$ is a metric space and the corresponding product topology is defined on $\displaystyle \mathbb{X}\times\mathbb{X}$. Prove that the metric d is continuous.
You have to show that if $\displaystyle (x,y)$ is close to $\displaystyle (x_0,y_0)$ in $\displaystyle \mathbb{X}\times\mathbb{X}$, then $\displaystyle d(x,y)$ is close to $\displaystyle d(x_0,y_0)$. More precisely, you have to show that, given $\displaystyle (x_0,y_0)\in\mathbb{X}\times\mathbb{X}$ and $\displaystyle \varepsilon>0$, there exists $\displaystyle \delta>0$ such that if $\displaystyle d(x,x_0)<\delta$ and $\displaystyle d(y,y_0)<\delta$ then $\displaystyle |d(x,y) - d(x_0,y_0)|<\varepsilon$. Use the triangle inequality.