Question

(5) Let F be an ordered field, and assume that the statement (Vxe Ft)(3! ye Ft) (y = x) is true. that F has denote by Nx the unique dement of Ftwhace square is x. Prove that of ae Ft and be Ft then na -16 = 9-b We say square roots, and for xe We J Na +ND (D) Let {xn. my be a sequence of points Ft and let bett Assume that {xo la convergent to b. Use port (@) that {Nang to No. (hint: It helps that yne ZZ", vån + No > No. to help prove is convergent

Answer 1

as ordered t Solha Given f is field. Art Ft, } ! y Ef such that yean. (a) Now Let a eft and beft Then rajo (ta.J6) (da tro) (a +56) as f field, ratato Ef and Jatro to. tardo ra. Tatja Jo-Jo. Ja-Torb sa tub a trador Taubob [sincef raddb is commutative, vadba sora] a- b Jato Thus Ja-Joa arb vatro (Proved)

(6) Since & ant converges to b Ero 7 mett such that for any lanabik E. Jo nyom Now , Isan-56 ancb Janaro I (by using & lan-bl 1 Jautro) laurbl <a [an eft, so daneft and SO Theret < { db уъ by , Jant67 db ie, tnym Santurto at . E othrom I datu-bike, turm. This shows that is & vanz converges nal convergent and to sb.