The standard weighted metric on a countable product of bounded complete metric spaces is complete

Given: The objects, hypotheses, and choice principles stated above.

The product set. Let $I$ be a set and let $X_i$ be a set for each $i\in I$. The product is ${\prod }_{i\in I}{X}_i:=\{x:x\text{ is a function with domain }I\text{ and }x(i)\in X_i\text{ for every }i\in I\},$ and we write $x_i:=x(i)$, the $i$-th coordinate of $x$. Two elements of the product are equal exactly when they agree at every index, functions being equal when they have the same domain and the same values. For $j\in I$ the $j$-th projection is ${\pi }_j:{\prod }_{i\in I}{X}_i\to X_j,{\pi }_j(x):=x_j\mathrm{.}$. The product topology ${\mathcal{T}}^{\mathrm{\Pi }}$ on ${\prod }_i{X}_i$ is the initial topology of the projections: the topology generated by the subbasis $\{{\pi }_i^{-1}[U]:i\in I,U\in {\mathcal{T}}_i\}$. Finite intersections of subbasic sets form a basis for it, and they are exactly the boxes ${\prod }_{i\in I}{U}_i$ with every $U_i$ open in $X_i$ and $U_i=X_i$ for all but finitely many $i$. (The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).

Let $(X,d)$ be a metric space (def-metric-space). $(X,d)$ is complete if every Cauchy sequence in $(X,d)$ converges to a point of $X$; a subset $A\subseteq X$ is called complete when the metric subspace $(A,d_A)$ is complete. (Complete metric space: every Cauchy sequence converges in the space).

Throughout, $\mathbb{R}$ is the complete ordered field (def-real-numbers) and a sequence of reals is a function $a:\mathbb{N}\to \mathbb{R}$ (def-sequence), written $(a_k)$; recall that $\mathbb{N}$ contains $0$. The sequence of partial sums of a sequence $(a_k)$ of reals is $s_n:={\sum }_{k<n}{a}_k$, so that $s_0=0$ and $s_{n+1}=s_n+a_n$; the series $\sum a_k$ converges when $(s_n)$ converges, and its sum is then that limit. (Series, partial sums, convergence and the sum, divergence, and the tail series).

Let $r\in \mathbb{R}$ and let $r^k$ be the integer power (def-integer-power), so that $r^0=1$ for every $r$, including $r=0$. 1. If $\mathrm{\mid }r\mathrm{\mid }<1$ then the series $\sum r^k$ converges (def-series) and ${\sum }_{k=0}^{\mathrm{\infty }}{r}^k=\frac{1}{1-r}\mathrm{.}$ 2. If $\mathrm{\mid }r\mathrm{\mid }\ge 1$ then $\sum r^k$ diverges. The series starts at $k=0$ and its first term is $r^0=1$; in particular ${\sum }_{k=0}^{\mathrm{\infty }}{2}^{-k}=2$, while the series starting at $k=1$ sums to $1$. Which starting index is meant has to be said, and it is said here. (For $\mathrm{\mid }r\mathrm{\mid }<1$, ${\sum }_{k\ge 0}{r}^k=1\mathrm{/}(1-r)$, and for $\mathrm{\mid }r\mathrm{\mid }\ge 1$ the series diverges).