= Solution
Let $(x^{(n)})$ be <Cauchy> for $d$, and fix a coordinate $i$. Every nonnegative summand is bounded by the whole sum:
$$
2^{-i}\phi\!\left(d_i(x_i^{(m)},x_i^{(n)})\right)\leq d(x^{(m)},x^{(n)}).
$$
Given $\varepsilon>0$, choose $N$ such that the right-hand side is less than $2^{-i}\varepsilon/(1+\varepsilon)$ for all $m,n\geq N$. Then strict monotonicity of $\phi$ implies $d_i(x_i^{(m)},x_i^{(n)})<\varepsilon$. Hence \b[every coordinate sequence is <Cauchy> in its original metric $d_i$], with the index bound allowed to depend on the fixed coordinate.
Back to article page
