Product space is completed with metric $d(x,y):=\sum_{i\in\mathbb{N}}\min\{2^{-i},d_i(x_i,y_i)\}.$

Question

We have the completed metric spaces $((S_i,d_i))_{i\in\mathbb{N}}$ and $S:=\Pi_{i\in\mathbb{N}}S_i$ with the metric $d(x,y):=\sum_{i\in\mathbb{N}}\min\{2^{-i},d_i(x_i,y_i)\}.$ How to show, that $(S,d)$ is also completed. Attempt: I want to show, that if $(x_n)_{n\in\mathbb{N}}$ is a Cauchy-Sequence in $S$, then $(x_{n_{i}})$ is a Cauchy-Sequence in $S_i, \ i\in\mathbb{N}.$ Then we would know, that $(x_{n_{i}})$ converges and I can set the limit $x$.

Answer 1

Each term in the infinite sum is less than or equal to the infinite sum. That is all you need to prove that the i-th coordinates form a Cauchy sequence for each i.