compact set in R = closed & bounded set in R
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Consider the set S = {1/n | n ∉ N} U {1/n + 1/m | n,m ∉ N} U {0} [N = set of natural numbers]
Clearly, this set is bounded.
Limit points of S are 0, 1, 1/2, 1/3, 1/4,..............,1/n,.......- which are countably infinite.
Also all the limit points are elements of S => S is closed.
Hence, S = {1/n | n ∉ N} U {1/n + 1/m | n,m ∉ N} U {0} is a compact set in R with countably infinite limit points.
Clearly, this set is bounded.
Limit points of S are 0, 1, 1/2, 1/3, 1/4,..............,1/n,.......- which are countably infinite.
Also all the limit points are elements of S => S is closed.
Hence, S = {1/n | n ∉ N} U {1/n + 1/m | n,m ∉ N} U {0} is a compact set in R with countably infinite limit points.