Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $ f(x)\leq f(x+\frac{1}{n} )$ for all $x\in \mathbb{R}$ and $n\geq 1$ . Prove that $f$ is non-decreasing .
I realy don't have any ideas. My first try was that by archemddian and well ordering property there exists a smallest $n$ such that $n(y-x)\geq 1 $ where it was chosen such that $y>x$ . Now if $y=x+\frac{1}{n} $ Then $f(y)=f(x+\frac{1}{n}) >f(x) $ . Now if $y>x+\frac{1}{n} $ i have no idea how to proceed .
So provide a solution . Thank you .