I've tried this question with Bing Chat (GPT-4) and Bard:
> A total order requires a This is the result Bing Chat gave me about a week ago (although I've just tried it a few times now, and all of the answers are considerably worse):
'A Cauchy sequence is a sequence of real numbers such that for any positive real number ε, there is a positive integer N such that for all natural numbers m and n greater than N, the absolute value of the difference between the m-th and n-th terms of the sequence is less than ε.
One way to define a This definition satisfies the properties of reflexivity (x Here is Bard:
'Let (x n) and (y n) be Cauchy sequences of real numbers. We define x n ≤ y n if and only if there exists a constant C such that ∣x n − y n∣ It is easy to verify that this definition satisfies the reflexivity, transitivity, antisymmetry, and strong connectedness properties of a total order.
Here is a more intuitive explanation of this definition. If x n ≤ y n, then the sequence (x n) is "eventually dominated" by the sequence (y n). In other words, there comes a point after which the terms of (x n) are always smaller than the corresponding terms of (y n).'
Second draft:
'Let (x n) and (y n) be two Cauchy sequences of real numbers. We say that (x n)≤(y n) if and only if for all ε>0, there exists an N such that for all n>N, we have ∣x n − y n∣Third draft is substantially similar to the second, just using a >= instead of > N. So so far this seems worse than GPT.