lean_workbook_plus_65318

Unfortunately, such a sequence cannot exist! Define $f(x) = \sqrt{x - \dfrac {1} {x}}$ for $x\geq 1$ . It is easily seen that $f(x) < x$ and $a_{n+1} = f(a_n)$ .\n\nIf such a sequence existed, then it would be decreasing and lower bounded by $0$ , so there will exist $a = \lim_{n\to \infty} = \inf a_n \geq 0$ .\n\nBut then $a$ would be a root for $x^3-x^2+1=0$ , which only has one negative real root, contradiction.