When I said when the sequence could decrease I was aiming to account for the cases where
an+1−an>1 (and aimed to note that tho was irrelevant).
(Reading through this part does seem horribly sloppy!) By successive terms equaling integers I meant that the difference between n and
an was always as integer. I lack the tools to explain what I mean rigorously but I thought the general idea seemed fairly intuitive:
Consider the values of the sequence where
bn supposedly "passes by" an integer r. As the sequence can only ever increase by fractions (bounded by an upper-bound that is inversely proportional to
an as n approaches infinity - and also fractions) this would require successive terms of
bn to equal two rational numbers above and below which would, in turn, imply that
gcd(an,n)<an (not co-prime, my bad!) for successive terms of n as n moves past r. This is a contradiction as, to "pass by" the integer would clearly require divisibility for one of the adjacent terms (i.e. r).
I'll attempt to formalise the last bit: clearly
an=an+1 as it "passes by" the value of r. So let
an=an+1=x. This implies
xn<r<xn+1 which implies
rx is not an integer (as its between n and n+1) therefore the inequality is not strict and r does infact equal either of the two (i.e. is in
bn). This is contrary to our assumption. Therefore no such r exists. QE"mother-****ing"D.