Question

(5) Here is a fascinating equivalence for being a complete metric space that we will use later. Let (X,d) be a metric space.

Answer 1

det (x, d) id cemplete Condider a bamily o Nen empty clos ed Subs etd Fo, Fy Fa X Such that im and Fo 2 Fi Fa2 - point in Fn arbit lady OA each m , bick an 2 m m N then Xn m ae im FN Say So o (nm) <diam FN diam F E arge Chersing very So Szn3 id caucny Sequence et im nN nE Fn So F Centaind at Least ene poiut aldo d1 3) diam Fn0 3 is in F

Cemarerdy we wiu Snow xis complete if Statea Conditiens aue Satis fiedl in X Let ny a Comcry Sequence Cendider Fn= nn+ 9 Then Fo 2 F? F2 Choose N do that d(nxm )<E EO) -3E, n>N > diam n nt -Siam Fn E nN diam Fun with Xo3= FonFinE o iin Fn do d(o %n)diomFno So