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,

a(n) = a(n-1) + b(n-1)

b(n) = a(n) + a(n-1)

Note that b(n) can be written

b(n) = 2a(n-1) + b(n-1)

and in the limit

Consider the ratio

And subsitute for b(n) and a(n).

Cross multiply to get

b(n)a(n-1) + b(n)b(n-1) = 2a(n)a(n-1) + a(n)b(n-1)

Divide both sides by a(n)a(n-1)

and use the limit equality

to simplify. The result is