2. Which of the following subset(o) of R (with the Buclidean metrie) is/are cloe

Question

2. Which of the following subset(o) of R (with the Buclidean metrie) is/are cloed? 2. 3. Which of the following sequence(s) is/are convergent? A. (cos(n)neN, B. (arctan(n)eC. ((In()))ne 3. 4. Which of the following metric space(s) (E, d) is/are complete? A. E = { 1, 2, 3, 4, 5} with d(z, y)-|e-y|. B. E-Q with d the discrete metric. C. E-R2 with d the Euclidean metric. 5. Given an example of each of the following (a) A collection (Un)neN of open subsets of R (with the usual metric) whose intersection U = nneNU, is neither open nor closed. Un = (b) A sequence of real nmumbers (an) that has two convergent subsequences (on) and (am) whose limits are different. an =

Explanation / Answer

2.A {0} is open as well as closed i.e clopen.

2.B  [0,1) is neither closed nor open.

It is not open because there is no >0 >0 such that (0,0+)[0,1)(0,0+)[0,1).

It is not closed because 11 is a limit point of the set which is not contained in it.

2.C (-,1] is closed .

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