Finding the limit of a sequence If $x_1$ and $y_1$ be positive numbers, $x_{n+1}$ $=$ $1\over 2$$(x_n+y_n)$ and $2\over y_{n+1}$ $=$ $1\over x_n$ $+$ $1\over y_n$ $\forall$ $n$ $\ge$ $1$. Show that the sequences $x_n$ and $y_n$ are monotonic and converge to a common limit $l$ where $l^2$ $=$ $x_1y_1$.
I proved that these two sequences are monotone, bounded and hence convergent. But I can't find that unique limit. I've been trying it with telescopic sum, but it doesn't help at all. 
Also, I failed to establish a good relation between $x_{n+1}$ and $x_n$ or $y_{n+1}$ and $y_n$.
Please help me to find that limit.

Hint:
$$\require{cancel}
x_{n+1} y_{n+1} = \frac{x_n+y_n}{2}\cdot\frac{2}{\cfrac{1}{x_n}+\cfrac{1}{y_n}}=\frac{\cancel{x_n+y_n}}{\bcancel{2}}\cdot\frac{\bcancel{2} \,x_n y_n}{\cancel{x_n+y_n}} = x_n y_n
$$
