213 



We can find, wlialever be «^0, an inleger k so that: 



[«4+1 i + ... + \ak+i,\ <-; 



then we liave also l^— s*| <— and |Q|n=|at^i <„_^)t + ••• + «» ^ | <C — 

 P—a^ t„ +...+ akt„^ic+ï — Ö, (^,+... + ^») +... + ai(6, + ...-|-è„_A:-|-i) 



= (6,i-i, + ... + ^,.-i) . («, + «2 + --+«it) + V,+^"-l^ + -.- + ^i-i+l «fc- 



^ ^..-Jl S/t + ii als fl = ft,, «1 + ... f K-k+1 sic 



We can find ft so that \b„—k+i\ <C.wt- if « ]> M ; then we have 



f 



also I /^ j <C "^ 'f " !> f*- 



Ö 



Now Sk tn-k = «<«-/t — (S — Sk) . In-k- 



B 



Since |(s— s;t) . <ii— ;t| <Cö (^*^® above), we have if n'^ii: 



Ó 



2e e 



\P—s . t„^k\ <C w ^"d' since \Q\ <C w '^"'^ "'" = -^^ + ^ • 



o o 



\Wu — S . tn-k\ <C«. 



Hence we see that it is possible to calculate, whatever be a ^ 0, 

 an integer (i which satisfies the conditions. 



Proof of theorem 5. 

 We have: 



n 1 »i '2 n— 1 ' ' n I ^ ' 



Hence, if 1 <^ k. <^ 7i -. 



n '- 1 H 'ft n— it+lJ ' "^ t+l H— it ' ' n 1 ^ 



Let ^'(p+i) <i and |*,|<«. 



Whatever be e ^ we can find an integer k so that : 



e 



\ak+i:\ +• •■ + \ak+p\ <Yt- 



ft 

 Then \s — sk\<C — and a fortiori: 

 o 



