Converging Subsequences: Finding a Sequence for All Integers

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SUMMARY

The discussion centers on constructing a sequence that includes subsequences converging to every integer, both positive and negative. The proposed sequence is $$0,-1,0,1,-2,-1,0,1,2,-3,-2,-1,0,1,2,3,\ldots$$ which effectively includes every integer infinitely. This sequence allows for the selection of constant subsequences that converge to any specified integer, addressing the initial challenge of including negative integers alongside positive ones.

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Carla1985
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I'm trying to find a sequence that has subsequences that converge to every integer. The question before that was the same but just for the positive integers, for which i gave {1,1,2,1,2,3...} but I'm struggling to include the negatives. Thanks
 
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Carla1985 said:
I'm trying to find a sequence that has subsequences that converge to every integer. The question before that was the same but just for the positive integers, for which i gave {1,1,2,1,2,3...} but I'm struggling to include the negatives. Thanks

Choose for example $$0,-1,0,1,-2,-1,0,1,2,-3,-2,-1,0,1,2,3,\ldots$$ and so on. Notice that every integer appears infinitely many times: we can choose a subsequence that converges to a given integer. Besides, that subsequence is constant.
 
That's fab, thank you! :)
 

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