If {x} is a sequence of rationals, I understand every real number will be a limit point. However, sequences have an order to them, right? So if this sequence of all rationals is monotonically increasing, then it will converge to infinite and all subsequences will have to converge to infinite. If it is monotonically decreasing, then it will converge to negative infinite and all subsequences will converge to negative infinite. How can the subsequence converge to a real number such as 1, when there exists a number x s.t. |x-1| > ε for any ε>0, such as when x = 1.1, since 1.1 is rational.(adsbygoogle = window.adsbygoogle || []).push({});

Can we ignore all values greater than one when constructing the subsequence? Then if I had a sequence from [0,-5) which converges to -5, can't I get subsequences with multiple subsequential limits, specifically every real number between [0,-5]. Then the upper limit of the sequence is -5, since this is convergence point of the sequence, but the supremum of subsequential limits is 0, since every real will be a subsequential limit, and 0 is the greatest number of the set.

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# In a sequence of all rationals, why is every real number a subsequential limit?

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