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If S is a subset of X,a metric space, I have always assumed that the definition of a limit point of S, say x, was that there a sequence in S converging to x. Therefore if x is in S, it is a limit point. That is, every member of S is a limit point of S, just by taking the sequence x,x,x,x,x,..... However on the wiki page, there is no statement that every member of a set is a limit point of the set, which is what I would expect.
Is this a 'correct' definition?
Is this a 'correct' definition?