Question

prove that the union of two bounded sets is a bounded set
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Answer 1

Answer:

Prove that the union of two totally bounded subsets of a metric space X is totally bounded. ... The set F ∪ G is finite, and its ϵ-neighbourhood contains A ∪ B: indeed, if x ∈ A ∪ B, then either x ∈ A or x ∈ B, and in both cases there is a point y ∈ F ∪ G at a distance < ϵ from x.