244 ARITHMETIC 4.2.4 Proof. Consider the (Web site construction) functions Q&s)
244 ARITHMETIC 4.2.4 Proof. Consider the functions Q&s) and R&s) defined by (ll), and let &m(t), t 5 r. (12) Sm(t) = &m(t) + &n(t), t > r. We will prove the lemma by induction on m. First note that &r(s) = (1 + (s - 1) + (r -10)/9)/s = 1 + (T -10)/9s, and RI(s) = (r -s)/s. From (8) we find that IPr(lOns) -Sr(s)/ = O(n)/lO ; this establishes the lemma when m = 1. Now for m > 1, we have and we want to approximate this quantity. By induction, the difference (13) is less than qc when 1 5 q 5 10 and j > N,-l(~). Since Sm-l(t) is continuous, it is a Riemann-integrable function; and the difference c s-,(~) -&1(t)dtl (14) lOj