← Back to Home

Order Relation on ℝ

Understanding how order is defined and preserved in ℝ

Well-Definedness of Order

If (aₖ) ~ (aₖ') and (bₖ) ~ (bₖ'), and aₖ < bₖ eventually, then aₖ' < bₖ' eventually.

0.511.52[aₖ] → 1.5[bₖ] → 0.8k (sequence index)(aₖ)(aₖ')(bₖ)(bₖ')

Key observation

Both green sequences converge to 1.5, both red sequences converge to 0.8. Since 1.5 > 0.8, eventually ALL green points are above ALL red points, regardless of which representative we choose.