= Countable product metric
{title2=$D(x,y)=\sum_{i\geq1}2^{-i}\frac{d_i(x_i,y_i)}{1+d_i(x_i,y_i)}$}
The displayed <metric> on a countable <Cartesian product> of <metric spaces> combines the <bounded metric transform> with summable positive weights. It is finite even when individual factor metrics are unbounded. The triangle inequality follows from subadditivity of $t/(1+t)$ and summation. Zero distance forces equality in each coordinate.
Convergence in this metric is exactly coordinatewise convergence. One term is bounded by the sum, so $D(x^{(n)},x)\to0$ implies each $d_i(x_i^{(n)},x_i)\to0$. Conversely, choose a finite head whose omitted weight sum is small; convergence in those finitely many coordinates controls the head uniformly, while boundedness of the transformed distances controls the entire tail. Thus it induces the countable <product topology>. Arbitrary positive summable weights work by the same proof.
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