Workshop Exercises 3.
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1.
Give an example of a NON-CONVERGENT sequence such that its limitpoint set contains only the number .
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2.
Give an example of an UNBOUNDED sequence such that its limitpoint set contains only the numbers . the elements and .
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3.
Show that the set is not closed.
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4.
Give an example of a sequence such that their limitpoints are exactly the prime numbers.
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5.
Let be closed sets in . Is it true that is always closed?
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6.
Let be a closed set in . Let . Show that there exists some such that the intersection of and the open interval is empty.
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7.
Let be a closed set . Let . Show that . More challenging version: Show that there exists such that .
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8.
Even more challenging version. Let and be two disjoint sets in . Prove that . Finally, show that there exists and such that