Consider $\displaystyle f_\lambda (x) = \lambda x(1 - x) $ for $\displaystyle x, \lambda \in \mathbb{R}.$

1) Show that $\displaystyle K_\lambda : = \{ x \in \mathbb{R}:$ the sequence $\displaystyle x, f(x), f(f(x)), \ldots $ is bounded $\displaystyle \}$ is always compact.

2) For which values $\displaystyle \lambda > 0$ is $\displaystyle K_\lambda$ connected?