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

Let $c\in {ℝ}_{+}$ 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 $x_0\in {ℝ}_{+}$. Let $u\in {ℝ}_{+}$, $d=c\cdot u^2$, $y_0=u{x}_0$ 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

$y_n=u{x}_n$

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

Create a solution