Determine with proof if the set 1 1 12 12 13 13 is compactS

Determine (with proof) if the set {-1, 1, -1/2, 1/2, -1/3, 1/3, ...} is compact.

Solution

A set of real number S is called a compact set if every sequence of its sub sequence will converge to an element which should be present in the given set S.

Now for real numbers its range from (- infinity ,+ infinity) .

hence all the set of real number must be converge to the element of Real number. Since real number set contain all types and every possible number.

Hence Compact set is defined as the Set S of real number if it is closed and bounded.

The given set is closed but it is not bounded .

So the given example is not an example of compact set

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