A show that d is a metric b show that d metrizes the

Question

If (Si , di ) are metric spaces for i ∈ I , where I is a finite set, then on the Cartesian product S = ITi ∈I Si let d(x, y) = },i di (xi , yi ).

(a) Show that d is a metric.

(b) Show that d metrizes the product of the di topologies.

(c) Show that (S, d) is complete if and only if all the (Si , di ) are complete.