Suppose that $f$ is a differentiable function and $p$ is a fixed point of $f$ such that $|f′(p)|<1$.
Let $K$ be the maximal interval about $p$ in which all points tend asymptotically to $p$ under $f$, i.e., K is the connected component of $\{x :f^n(x) \to p$ as $n \to \infty\}$ which contains p. $K$ is known as the immediate basin of attraction of $p$.
How to prove that $f(K) \subset K$?
ps. $f^n$ is the n-time composition of $f$ whit itself