Constant function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function such that $\left | f(x)-f(y) \right |\leq \left | x -y \right |^2 , \;\; \forall x, y \in \mathbb{R}$. Prove that $f$ is constant.

Solution

Is completeness theorem satisfied?

Does the ordered field of the rational functions satisfy the completeness theorem: "all non - empty sets have a supremum")

Solution