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In the chapter on nets in Bredon's Topology and Geometry, there is a theorem that every net has a universal subnet. There is also an exercise that asks you to prove that a sequence is a universal net if and only if it is eventually constant. These two statements seem to imply that every sequence has a subnet (subsequence) that is eventually constant, which appears to be trivially false (consider, for example, the sequence x_n = n). Where am I going wrong?

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    $\begingroup$ A subnet of a sequence is not necessarily a sequence $\endgroup$ Commented Jun 11 at 11:57
  • $\begingroup$ What is a universal subnet? $\endgroup$
    – Paul Frost
    Commented Jun 11 at 21:59
  • $\begingroup$ A net is universal if for any subset of the ambient space, the net is eventually contained in either that subset or its complement. It gives a neat characterization of compactness: a space is compact if and only if every universal net converges. $\endgroup$ Commented Jun 11 at 22:35

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