Heron's method/Square root 3/Square root 1 divided by 3/Exercise

Let ${\left(x_n\right)}_{n\in \mathbb{N}}$ denote the Heron-sequence for the computation of $\sqrt{3}$ with initial value $x_0=1$ and let ${\left(y_n\right)}_{n\in \mathbb{N}}$ be the Heron-sequence for the computation of $\sqrt{\frac{1}{3}}$ with initial value $y_0=1$.

1. Compute $x_1$ and $x_2$.
2. Compute $y_1$ and $y_2$.
3. Compute $x_0\cdot y_0,x_1\cdot y_1$ and $x_2\cdot y_2$.
4. Does the product sequence $z_n=x_n\cdot y_n$ converge within the rational numbers?

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