4. Show that is not equivalent to the usual metric space on R. 5. Is & topologically equivalent to the usual metric space on R.

Trigonometry (MindTap Course List)
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Author:Ron Larson
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Chapter2: Analytic Trigonometry
Section2.3: Solving Trigonometric Equations
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topology campact metric part 4 5
Exercise 1. Let z ad y be two reals. Set for z and y,
6(x, y) = tan¹r-tan-¹ y.
1. Show that is a metric on R.
2. Is R bounded with respect to this metric?
3. Compare the open balls B(0,2), B(0,4), and B(1,4). What do you observe?
4. Show that is not equivalent to the usual metric space on R.
5. Is á topologically equivalent to the usual metric space on R.
6. Show that R is not sequentially compact with respect to the metric 6.
7. Deduce.
Transcribed Image Text:Exercise 1. Let z ad y be two reals. Set for z and y, 6(x, y) = tan¹r-tan-¹ y. 1. Show that is a metric on R. 2. Is R bounded with respect to this metric? 3. Compare the open balls B(0,2), B(0,4), and B(1,4). What do you observe? 4. Show that is not equivalent to the usual metric space on R. 5. Is á topologically equivalent to the usual metric space on R. 6. Show that R is not sequentially compact with respect to the metric 6. 7. Deduce.
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