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I understand that this sequence has sup=1 and inf=0, but since I do not know what the sequence will look like as n approaches infinity, how do I know lim sup=1 and lim inf=1?

For example, if the sequence is non-decreasing, then can I necessarily say that lim inf=0. I know that the inf will be 0, but lim inf cannot be approaching this number since the rational numbers are increasing. Tell me where my thinking is wrong.