/sci/ - Science & Math

What's the purpose of taking the supremum of the infimum here in equation A1.5? This is apparently supposed to extend arithmetic to infinite decimals.

(D is the set of finite decimals)

Could someone explain how to prove that the infimum and supremum of the function in equation A1.5 exist? Like, how do we know that f([x_1]_l, ..., [x_n]_l) has a lower bound and greatest lower bound for l<=-k, and how do we know these infima have a least upper bound?

[D is the set of finite decimals by the way].

Can anyone prove proposition A1.6? (D is the set of finite decimals).

Can someone explain proposition A1.6 here? Apparently it's used to define arithmetic between real numbers as infinite decimals.