Bing follow-up: 'Can you prove those properties?

'Sure! Here's a proof for each property:

Reflexivity: For any real number x represented by a Cauchy sequence (xn), we have that for any positive real number there exists a positive integer N such that for all natural numbers n greater than N, xn - xn = 0 <= ε. Thus, x <= x.

Transitivity: Suppose x <= y and y <= z for real numbers x,y,z represented by Cauchy sequences (xn), (yn), and (zn) respectively. Then for any positive real number there exist positive integers N1 and N2 such that for all natural numbers n greater than N1 and m greater than N2, xn - yn <= ε/2 and ym - zm <= ε/2. Let N
max(N1,N2). Then for all natural numbers n greater than N, we have xn - zn = (xn - yn) + (yn - zn) <= ε/2 + ε/2 = Thus, x <= z.

Anti-symmetry: Suppose x <= y and y <= x for real numbers x,y represented by Cauchy sequences (xn) and (yn) respectively. Then for any positive real number ε, there exist positive integers N1 and N2 such that for all natural numbers n greater than N1 and m greater than N2, xn - yn <= ε/2 and ym - xm <= ε/2. Let N = max(N1,N2). Then for all natural numbers n greater than N, we have |xn - yn| = |I(xn - yn) + = I(xn - yn) + (ym - xm)| <= |xn-yn| + lym-xml<=ε/2+ε/2=ε. Since this holds true for any positive real number it follows that the sequence (|xn-yn|) converges to 0 which implies that the two Cauchy sequences represent the same real number i.e., x=y.

Strong connectedness: For any two real numbers represented by Cauchy sequences (xn) and (yn), either there exists a positive integer M such that xn- yn<=0 or there exists a positive integer M such that yn-xn<=0. In the first case we have that x<=y while in the second case we have y<=x.

Hope this helps!'

Bard second try:

Prompt: 'A total order requires a <= relation which must be reflexive, transitive, anti-symmetric, and strongly connected. On Cauchy sequences, we define a sequence {x_n} <= {y_n} if for any ε > 0, there exists N such that if n > N, then x_n - y_n <= ε. Prove that this definition of <= on Cauchy sequences is a total order.'

Bard:

'Sure. Here is a proof that the definition of <= on Cauchy sequences is a total order:

Reflexivity: Let x n be a Cauchy sequence. Then for any ε>0, there exists N such that if n>N, then x n − x n < ε. Therefore, x n <= x n.

Transitivity: Let x n and y n be Cauchy sequences. Then for any ε>0, there exists N such that if n>N, then x n − y n < ε and y n − z n < ε. Therefore, x n − z n < 2ε.

Anti-symmetry: Let x n and y n be Cauchy sequences. If x n <= y n and y n <= x n, then for any ε>0, there exists N such that if n>N, then x n − y n < ε and y n − x n < ε. Therefore, x n = y n.

Strong connectivity: Let x n and y n be Cauchy sequences. Then for any ε>0, there exists N such that if n>N, then x n − y n<ε or y n − x n < ε. Therefore, x n and y n are connected.

Therefore, the definition of <= on Cauchy sequences is a total order.'

Reflexivity and transitivity were right for both, Bing was dead wrong about anti-symmetry, Bard gave enough to prove anti-symmetry for a reasonable definition of equality of Cauchy sequences, and neither got a proof of strong connectedness right, they essentially assert it without proving it. But that is substantially more difficult than the other properties to prove.