(related to Proposition: Isometry is Injective)
Let \((X,d_X)\) and \((Y,d_Y)\) be metric spaces and let \(a,b\in X\) with \(a\neq b\). Becasue \(d_X\) is a metric, we have \(d_X(a,b)\neq 0\). Since \(f:X\mapsto Y\) is an isometry by hypothesis, we have \[d_Y(f(a),f(b))=d_X(a,b)\neq 0\] for some \(f(a),f(b)\in Y\). Because also \(d_Y\) is a metric, we must have \(f(a)\neq f(b)\). We have shown \[a\neq b\Longrightarrow f(a)\neq f(b).\] Thus, \(f\) is injective.