Get $\{(X_i,d_i):i\in \mathbb {N}\}$ a family metrics spaces. Get $X=\{\{x_i\}_{i\in \mathbb{N}}:x_i\in X_i\}$ Proof that $(x,d)$ is a metric space.

Question

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Get $\{(X_i,d_i):i\in \mathbb {N}\}$ a family of metrics spaces and get. $$X=\{\{x_i\}_{i\in \mathbb{N}}:x_i\in X_i\}$$

Define $$d:X \text{x} X \to \mathbb{R_{>0}} $$ For, $$d(\{x_i\}_{i\in \mathbb{N}},\{y_i\}_{i\in \mathbb{N}})=sup\{d_1(x_i,y_i):i\in \mathbb{N}\}$$ Where, $$d_1(x_i,y_i)=min\{1,d_i(x_i,y_i)\}$$

Proof that $(x,d)$ is a metric space.