Let $\displaystyle (M_1,d_1),\ldots,(M_n,d_n),$ and $\displaystyle M=M_1\times\cdots\times M_n.$ Prove the following:

$\displaystyle 1_M : (M,d_e)\to(M,d_m)$ and $\displaystyle 1_M :M,d_e)\to(M,d_s)$ are homeomorphisms, where $\displaystyle d_e$ is the euclidian distance, $\displaystyle d_s=|x_1-y_1|+\cdots+|x_n-y_n|,$ and $\displaystyle d_m$ distance of the maximum.

$\displaystyle 1_M$ is supposed to be the identity function, but I'm confused how to work with.

$\displaystyle f( N,d)\to (M,d_e)$ is continuous iff $\displaystyle f_i( N,d)\to(M_i,d_i)$ is continuous for $\displaystyle 1\le i\le n.$

The left implication is easy because if each $\displaystyle f_i$ is continuous then clearly $\displaystyle f$ is continuous, but don't know how to prove the right implication.