Let (, ) be a metric space. For nonempty bounded subsets and let (,) inf(,) : and (,) sup(,) : . Now define the Hausdorff metric as (, ) max(, ), (, ). Note: can be defined for arbitrary nonempty subsets if we allow the extended reals. a) Let P() be the set of bounded nonempty subsets. Prove that (, ) is a so-called pseudometric space: satisfies the metric properties (i), (iii), (iv), and further (, ) = 0 for all . b) Show by example that itself is not symmetric, that is (, ) (, ). c) Find a metric space and two different nonempty bounded subsets and such that (, ) = 0