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  3. 1. let ?: r2 ×r2 ?r be given by ?((x1,y1),(x2,y2)

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

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Advance Math

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

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

(b) In (R2,?), sketch the open ball with center (0,0) and radius1. 2. Let {xn} be a sequence in a metric space (X,?). Prove that ifxn ? a and xn ? b for some a,b ? X, then a = b.

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

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