Show that the map d: R x R R defined by d(x, y) = ||x-y||/1+||x-y||...

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Show that the map d: R" x R" → R defined by d(x, y) = √||x-y||/1+√||x-y|| ,is a metric, where || || denotes the Euclidean norm on R". (ii) Let d be the above metric. Show that not all closed and bounded subsets of (R", d) are compact. (iii) Does the above phenomenon provide a counterexample to the Heine-Borel theorem? r-yll 1+ ||z-y|| Show that the map d: R" x R" → R defined by d(x, y) = √||x-y||/1+√||x-y|| ,is a metric, where || || denotes the Euclidean norm on R". (ii) Let d be the above metric. Show that not all closed and bounded subsets of (R", d) are compact. (iii) Does the above phenomenon provide a counterexample to the Heine-Borel theorem? r-yll 1+ ||z-y||

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