Equivalence of Product Metric and $d_1$ Metric

Question

I'm trying to show that the product metric $d_{X \times Y}((x_1,y_1),(x_2,y_2))=\max{\{d_X(x_1,x_2),d_Y(y_1,y_2)\}}$ and the metric $d_1((x_1,y_1),(x_2,y_2))=d_X(x_1,x_2)+d_Y(y_1,Y_2)$ define equivalent metrics on $X \times Y$.

My definition of equivalence of metrics is that there exist constants $c$ and $c'$ such that $cd_1((x_1,y_1),(x_2,y_2)) \leq d_{X \times Y}((x_1,y_1),(x_2,y_2)) \leq c'd_1((x_1,y_1),(x_2,y_2))$.

I'm thinking that $c$ should be $\frac{1}{2}$ and $c'$ should be $2$. Is my thinking correct?

Answer 1

Clearly,$$\max\bigl\{d_X(x_1,x_2),d_Y(y_1,y_2)\bigr\}\leqslant d_X(x_1,x_2)+d_Y(y_1,y_2).$$On the other hand, since $d_X(x_1,x_2),d_Y(y_1,y_2)\leqslant\max\bigl\{d_X(x_1,x_2),d_Y(y_1,y_2)\bigr\}$,$$d_X(x_1,x_2)+d_Y(y_1,y_2)\leqslant2\max\bigl\{d_X(x_1,x_2),d_Y(y_1,y_2)\bigr\}.$$