This is a simplified version of robjohn's answer.
By the AM-Hm inequality we have $y_n \leq x_n$. Moreover
$$x_n = \frac{x_{n-1}+y_{n-1}}{2} \leq \frac{ x_{n-1}+x_{n-1}}{2}=x_{n-1}$$ and
$$y_n=\frac{2}{\frac{1}{x_{n-1}}+\frac{1}{y_{n-1}}}\geq\frac{2}{\frac{1}{y_{n-1}}+\frac{1}{y_{n-1}}}=y_{n-1}$$
This shows that $y_1 \leq y_2 \leq ..\leq y_n \leq ..\leq x_n \leq x_{n-1} \leq ... \leq x_1$$
Hence both sequences are monotonic and bounded thus convergent. Let
$$\lim x_n=l \,;\, \lim y_n =m \,.$$
Then taking the limit in $x_{n+1}=\frac{x_n+y_n}{2}$ we get $l=m$.