1. Let ρ: R2 ×R2 →R be given by ρ((x1,y1),(x2,y2)) = |x1 −x2|+|y1 −y2|. (a) Prove...

1. Let ρ: R2 ×R2 →R be given by ρ((x1,y1),(x2,y2)) = |x1 −x2|+|y1 −y2|.

(a) Prove that (R2,ρ) is a metric space.

(b) In (R2,ρ), sketch the open ball with center (0,0) and radius 1. 2. Let {xn} be a sequence in a metric space (X,ρ). Prove that if xn → a and xn → b for some a,b ∈ X, then a = b.

3. (Optional) Let (C[a,b],ρ) be the metric space discussed in example 10.6 on page 344 of Wade. If {fn} and f are in C[a,b], prove that fn → f with respect to ρ if and only if fn → f uniformly (in the sense of Chapter 7).

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Colby Messinger answered 1 year ago

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