Do these two sequences converge to same limit? $(x_n)_{n\geqslant1}$ and $(y_n)_{n\geqslant1}$ are two real sequences such that   $x_1 > 0$, $y_1 > 0 $, $x_{n+1} = \frac{1}{2}(x_n + y_n)$ and $\dfrac{2}{y_{n+1}} = \dfrac{1}{x_n} + \dfrac{1}{y_n}$.
Do these two sequences converge to the same limit? If yes, what is their limit?

if we divide 1st one by x_n
$$\frac{x_{n+1}}{x_n}=\frac{1}{2}\left(1+\frac{y_n}{x_n}\right)$$
then 2nd one multiply by y_n
$$\frac{y_n}{y_{n+1}} =\frac{1}{2}\left(1+\frac{y_n}{x_n}\right)$$
