Chapter 7.7, Problem 15E

(a)

To determine

To Calculate: The ${\left(\sqrt{2}-1\right)}^n=x_n+y_n\sqrt{2}$. If n is interger, and that there exist intergs $x_n$ and $y_n$, depeding on n.

(b)

To determine

To Calculate: The $\underset{x\to \infty }{\lim }{\left(\sqrt{2}-1\right)}^n=0$.

(c)

To determine

To Calculate: The $\sqrt{2}$ is rational and that (a) and (b) lead to a contradiction. For $n\ge 0.$