Limit of infinite sequence.

The sequence is for x>y>0

$\displaystyle

a_1 = x +y,

$

$\displaystyle

a_n = x + y - \frac{xy}{a_{n-1}}

$

Now what i have got is

$\displaystyle

a_n = \frac{x^{n+1}-y^{n+1}}{x^n - y^n}

$

How do i find the limit when n goes to infinity ?

Edit: I suspect it has somthing to do with (I dont know the correct english word) the rule that when:

$\displaystyle

(a_n) \leq (c_n) \leq (b_n)

$

$\displaystyle

\lim\limits_{n \to \infty} a_n = \lim\limits_{n \to \infty} b_n

$

Then:

$\displaystyle

\lim\limits_{n \to \infty} c_n = \lim\limits_{n \to \infty} a_n

$

But I am unable to construct two such limits.