Notice that the subsequences of even numbered terms and odd numbered terms are monotone and bounded. This gives a basis for proof that a limit exists.
To evaluate the limit,
Note that b(n) can be written
and in the limit
Consider the ratio
And subsitute for b(n) and a(n).
Cross multiply to get
Divide both sides by a(n)a(n-1)
and use the limit equality
to simplify. The result is
