Give examples of a nested sequence of closed intervals sothat a. no point lies i

Question

Give examples of a nested sequence of closed intervals sothat a. no point lies in all of them b. exactly one point lies in all of them c. the set of points lying in all of the closed intervals is aclosed interval d. the set of points lying in all of the closed intervals isan open interval Give examples of a nested sequence of closed intervals sothat a. no point lies in all of them b. exactly one point lies in all of them c. the set of points lying in all of the closed intervals is aclosed interval d. the set of points lying in all of the closed intervals isan open interval

Explanation / Answer

a. no point lies in all of them This is not possible. The nested intervalstheorem states that if each In is a closed and boundedinterval, say
b. exactly one point lies in all of them In = [-1/n,1/n] Then the intersection ofthe In is {0}, a singletonset.

c. the set of points lying in all of the closed intervals is aclosed interval     In  = [-1/n, 1+1/n]    Then the intersection ofthe In is [0,1] a closedinterval.

d. the set of points lying in all of the closed intervals is anopen interval   This is not possible since intersection of anycollection closed sets is again a closed set. Note the closedintervals are closed sets.    But an open interval is not a closed set. Hence theintersection of closed intervals cannot be an open interval. a. no point lies in all of them This is not possible. The nested intervalstheorem states that if each In is a closed and boundedinterval, say In =[an, bn] with an bn then under the assumption of nesting, the intersection ofthe In is not empty.
b. exactly one point lies in all of them In = [-1/n,1/n] Then the intersection ofthe In is {0}, a singletonset.

c. the set of points lying in all of the closed intervals is aclosed interval     In  = [-1/n, 1+1/n]    Then the intersection ofthe In is [0,1] a closedinterval.

d. the set of points lying in all of the closed intervals is anopen interval   This is not possible since intersection of anycollection closed sets is again a closed set. Note the closedintervals are closed sets.    But an open interval is not a closed set. Hence theintersection of closed intervals cannot be an open interval.

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