c22
Further, we have
α!+ιJ∕1 - bn+1 = a^+1(xn+1 + q(1)) - bn+1
≥ a^+1xn+1 - bn+1
> a~[ xn+1 — b1
> a~[ (æn+1 + ¢(1)) — &i
= a~[y1-b1.
It means that l(y1) ψ 1. It is a contradiction.
(2). If 1 < к < n + 1, then yk = xk~k ⅛ q(k'). Since l(xk~r) = к — 1, it implies
that
a~l^xk~k - bk~1 ≥ a^xk~1 - bk.
Hence we have
ak.1yk - bk^1 = a^1(xk~1 + q(k∖) - bk.1
= a^1xk~1 - bk.1 + a^l-1q(k)
> a[xk~k — bk + a[q(k)
≥ a[(xk~1 + q(k}) - bk
= akyk-bk.
It means that l(yk) ψ k. It is again a contradiction.
(3). If к = n + 1, then yn+k = x1 — q(∏). Notice that
O.nX b∏ ≥ ¼ι+l∙
We have
a∏yn+1-bn = ɑɪ(æ1 - q(n}) - bn
≥ dn+ιX1 - bn+ι -aτnq(π)
> aτn+1x1 - bn+1 - a^+1q(n)
= an+ι(χ1 - q(n∖) - bn+ι
= a^+1yn+1 - bn+1.