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P: 80
 Quote by lugita15 Yes, sorry. I think you may still be able use my suggestion to prove 2 implies 3 without using 1,4, or 5. On a seperate note, you can try proving 2 implies 5 instead (because you've already proven that 1,3,4, and 5 are equivalent, so the fact that 1 implies 2 and 2 implies 5 means that 2 is equivalent to the rest). One simple strategy is to try constructing a nested sequence of intervals whose lengths go to zero using the elements of a Cauchy sequence.
Even though I would like a more direct approach 2-5 will suffice.
Suppose that I want to prove that a Cauchy sequence x_n converges
How can I create a sequence of nested intervals whose lengths go to 0 when x_n is not necessarily monotonous?