# Closing lemma on the Interval

Let $f:$ $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ a $C^1$ diffeomorphism, $x\in\Omega(f)$.$\space$

How do i proove that $\forall\space\epsilon\gt0$, $\exists\space$ $g:$$\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ with $d$$f,g$$\lt\epsilon$ $\space$ such that $x\in Per(g)$

$\Omega(f)=$ { $y\in\\mathbb{R}\mid\space\forall\space U, y\in U,$ there is $n\ge1$ such that $f^n$$U$$\cap U\neq\emptyset$ }

The distance is given by the uniform convergence of the functions and their first derivative.

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Since your $f$ is a diffeo, it is monotone, and a wandering point of a monotone map on $\mathbb{R}$ is already either a fixed point or a point of period 2 (the latter can only happen if $f$ is decreasing).