Solution
= Solution
A base covers the space and refines intersections. On $X^k$, products of opens form a base. The weighted metric satisfies the metric axioms and has exactly this topology; convergence is coordinatewise. For $X^{\mathbb N}$ the metric $d(f,g)=\sum_{n\ge1}2^{-n}\min\{1,\rho(f(n),g(n))\}$ induces the stated <product topology>: finitely many coordinates control a basic neighbourhood and the tail is uniformly small. Thus <sequences> converge exactly coordinatewise.