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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