Sequence generated by arithmetic and harmonic mean converges to square root
Solution 1:
Let us check that $(x_n)_n$ is a convergent sequence. It is monotonously decreasing:
$$x_{n+1}=\frac{x_n+y_n}{2}<\frac{x_n+x_n}{2}=x_n$$
On the other hand the sequence is bounded from below: $x_{n}\ge 0$.
Therefore it converges to some limit $x=\lim_{n} x_n$ (monotone convergence theorem). Since $x_ny_n=ab$ for all $n$, also $(y_n)_n$ converges to some limit $y=\lim_n y_n$. Now the recurrence for $x_n$ implies
$$x=\frac{x+y}{2}$$
That is, $x=y$. Since $xy=ab$, we have $x=y=\sqrt{ab}$.