Google releases Bard to a limited number of users in the US and UK
I&#x27;ve tried this question with Bing Chat (GPT-4) and Bard:<p>&gt; A total order requires a &lt;= relation which must be reflexive, transitive, anti-symmetric, and strongly connected. Give a definition for &lt;= on real numbers defined using Cauchy sequences.<p>This is the result Bing Chat gave me about a week ago (although I&#x27;ve just tried it a few times now, and all of the answers are considerably worse):<p>&#x27;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.<p>One way to define a &lt;= relation on real numbers using Cauchy sequences is to say that for two real numbers x and y represented by Cauchy sequences (xn) and (yn), x &lt;= y if and only if for any positive real number there exists a positive integer N such that for all natural numbers n greater than N, xn - yn &lt;= ε.<p>This definition satisfies the properties of reflexivity (x &lt;= x), transitivity (if x &lt;= y and y &lt;= z then x &lt;= z), anti-symmetry (if x &lt;=y and y &lt;= then x = y) and strong connectedness (for any two real numbers x and y, either x &lt;=y or y &lt;= x).&#x27;<p>Here is Bard:<p>&#x27;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∣ &lt; C for all sufficiently large n.<p>It is easy to verify that this definition satisfies the reflexivity, transitivity, antisymmetry, and strong connectedness properties of a total order.<p>Here is a more intuitive explanation of this definition. If x n ≤ y n, then the sequence (x n) is &quot;eventually dominated&quot; 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).&#x27;<p>Second draft:<p>&#x27;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&gt;0, there exists an N such that for all n&gt;N, we have ∣x n − y n∣&lt;ε.&#x27;<p>Third draft is substantially similar to the second, just using a &gt;= instead of &gt; N. So so far this seems worse than GPT.