# Sard's theorem for orientation preserving diffeomorphism of the circle

thanks in advance for helping me. First I'll introduce some definitions:

(1) Suppose that $$f : \mathbb{S}^{1} \rightarrow \mathbb{S}^{1} = \mathbb{R} / \mathbb{Z}$$ is an orientation preserving diffeomorphism of the circle. Then $$f$$ may be lifted to a homeomorphism $$F:\mathbb{R} \rightarrow \mathbb{R}$$ of the real line, satistying $$F(x+m)=F(x)+m.$$ The rotation number $$\rho(f)$$ of $$f$$ is defined as the fractional part of $$\rho_{0}(f) = \lim_{n\to\infty} \frac{F^{n}(x)}{n}.$$ That is, $$\rho(f)$$ is in the unique number in $$[0,1)$$ for which $$\rho_{0}(f)-\rho(f)$$ is an integer. The rotational number exists (proved by Poincaré) and is independent of the starting point $$x$$.

(2) An orientation of a smooth m-dimensional manifold $$M$$ is a choice of orientation on every tangent space $$T_{x}M$$ satisfying the condition: for each $$x \in M$$ there is a neighborhood $$U \subset M$$ and an orientation-preserving diffeomorphism (i.e. a coordinate system) $$h$$ from $$U$$ to an open set in $$\mathbb{R}^{n}$$

(3) Let $$f$$ and $$g$$ be two k-times differentiable functions from $$\mathbb{R}$$ to $$\mathbb{R}$$. The $$C^{1}$$-distance between $$f$$ and $$g$$ is given by $$d_{1}(f,g)=\sup_{x\in \mathbb{R}}(|f(x)-g(x)|,|f^{(1)}(x)-g^{(1)}(x)|).$$

Now, my question: how to prove the following proposition using Sard's theorem?

Suppose $$f$$ is an orientation-preserving diffeomorphism of the circle with $$\rho(f) = 0$$. Then there is a $$C^{1}$$ (meaning continuously differentiable) diffeomorphism $$g$$ which is arbitrarily close to $$f$$ with respect to the $$C^{1}$$-distance and which has only isolated fixed points.