Question

For (a, b), (c, d) E R2, define if b d This defines a metric on R2. (You need not check this now.) (a) Plot a few points in the plane, and find the distances between them, as mea- sured by de. Choose cases that you think will be illuminating b) Explain why this metric could be called the "lift" metric. (Think about trav- elling between floors in a hotel.) (c) i. Draw the open ball, centre (3, 0), radius 10, in this metric space. Thern draw more open balls, still with centre (3,0), but with decreasing radii. ii. Draw the open ball, centre (0,3), radius 10, in this metric space. Then draw more open balls, still with centre (0, 3), but with decreasing radii. iii. Use the pictures of your open balls to characterise the convergent sequences in this metric space. (You will need to consider two cases.)

Answer 1

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