(a) It is well-defined as the product is finite. All properties including the triangle inequality are clearly satisfied (similar to the proof on page 122). Finally, $B_{\rho}(\mathbf{x},r)=\prod_{i=1}^{n}B_{d_{i}}(x_{i},r)$ is open in the product topology, and for a basis element $\prod_{i=1}^{n}B_{d_{i}}(x_{i},r_{i})$ for the product topology and any its point $\mathbf{y}$ , for each $i=1,\ldots,n$ , we can find $s_{i}$ such that $B_{d_{i}}(y_{i},s_{i})\subset B_{d_{i}}(x_{i},r_{i})$ , and, by letting $s=\min\{s_{1},\ldots,s_{n}\}$ , $\mathbf{y}\in B_{\rho}(\mathbf{y},s)=$ $\prod_{i=1}^{n}B_{d_{i}}(y_{i},s)\subset$ $\prod_{i=1}^{n}B_{d_{i}}(y_{i},s_{i})\subset$ $\prod_{i=1}^{n}B_{d_{i}}(x_{i},r_{i})$ .

(b) $\overline{d}_{i}$ is a metric on $X_{i}$ inducing the same topology as $d_{i}$ (Theorem 20.1), and, therefore, so is $\overline{d}_{i}/i$ . $D$ is well-defined as the set $\{\overline{d}_{i}(x_{i},y_{i})/i\}$ is bounded from above, and all the metric properties hold for $D$ (similar to Theorem 20.5). Further, again similar to Theorem 20.5, for $B_{D}(\mathbf{x},r)$ , $\prod_{i=1}^{N}B_{d_{i}}(x_{i},r)\times\mathbb{R}\times\mathbb{R}\times\cdots\subset B_{D}(\mathbf{x},r)$ if $\tfrac{1}{N}\le r$ , and for $U=\prod_{i\in\mathbb{Z}_{+}}U_{i}$ where $U_{i}=\mathbb{R}$ if $i>N$ , and any point $\mathbf{y}\in U$ , we can find $B_{d_{i}}(y_{i},s_{i})\subset U_{i}$ , $s_{i}\le1$ , for $i=1,\ldots,N$ , take $s=\min\{\tfrac{s_{1}}{1},\ldots,\tfrac{s_{N}}{N}\}$ , and then $\mathbf{y}\in B_{D}(\mathbf{y},s)\subset$ $B_{d_{i}}(y_{1},s_{1})\times\cdots\times B_{d_{N}}(y_{N},s_{N})\times\mathbb{R}\times\mathbb{R}\times\cdots\subset U$ .