User blog comment:Vel!/FGH Gripe/@comment-5982810-20150325004329/@comment-1605058-20150326141125

[NOTE: this comment is not meant to be a counterpoint to Sbiis's (1), because the scenario like the one I'm describing probably (but not certainly!) wouldn't happen in "natural" (whatever that means) FS systems]

It turns out that having b[n]_1 <= b[n]_2 for every ordinal b and natural n doesn't guarantee that for every a we have f_a(n) <= g_a(n) (using f and g like Sbiis did).

Let's take for every ordinal a2k. Let's take e0[n]_1=w^^(2n+1) and e0[n]_2=w^^(2n+2). Of course, a[n]_1 and a[n]_2 agree everywhere below e0, and e0[n]_1g_e0(n).