Heron's method/R/Dilation by square/Exercise

Let ${}c\in \mathbb {R} _{+}$ be a positive real number and let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be the Heron-sequence for the computation of ${}{\sqrt {c}}$ with the initial value ${}x_{0}\in \mathbb {R} _{+}$. Let ${}u\in \mathbb {R} _{+}$, ${}d=c\cdot u^{2}$, ${}y_{0}=ux_{0}$ and let ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ be the Heron-sequence for the computation of ${}{\sqrt {d}}$ with initial value ${}y_{0}$. Show that

${}y_{n}=ux_{n}\,$

holds for all ${}n\in \mathbb {N}$.

Create a solution