We can assume that b>a

 

Looks like the only conclusion is that the two sequences converge towards each other as we always have a_n<b_n<b_n-1. Thus b_n is a decreasing sequence bounded below by a^_n. Also a_n is an increasing sequence bounded above by b_n^.

 

A similar argument can be made for the case b<a.

 

I would also like to point out that inf+1  inf does not lead us to conclude that a_na_n-1 etc.as n goes to inf