It's linear for small redshifts only. In general it depends on what distance you mean, but it's never linear. This does not change the validity of your argument

It's linear for small redshifts only. In general it depends on what distance you mean, but it's never linear. This does not change the validity of your argument of course. Check https://en.m.wikipedia.org/wiki/Distance_measure https://en.m.wikipedia.org/wiki/Distance_measure