Solution 80Let the statement
ann−1→0 be
(⋆)(⋆)⇒ there must be an integer
m for which
am=m and
an>m<n. Suppose that for some
k there exists
i≥m such that
aii−1=k−1. If we are lucky and
ai=ai+ai then we have
ai(i+ai)−1=(k+1)−1 and we are done. On the other hand, if
ai increases, bearing
ai<i and
(⋆) in mind there will be
j>i such that
ajj−1=k−1, which brings us back to our hypothesis. However, by
(⋆) this can only occur finitely many times for any fixed
k, and hence
an will always contain a constant string which leads to our lucky case. Thus the induction is complete and the sequence
n−1 is a subsequence of
ann−1.
The use of the limit can perhaps be made clearer by noting that
(an) must contain arbitrarily long strings of repeated integers. This is easily shown: