If $\liminf_{x\to \infty}f(x)>M,$ for some $M>0$, where $f$ is continuous function on $\mathbb R$.
Does this imply that: there exist $x_{o}$ such that for all $x\geq x_{o}$ we have $f(x)>M$? If so, how I can prove it?
I also have another question, just to be sure: If $f(x)\leq g(x)$ for all $x\in \mathbb R$, both are continuous on $\mathbb R$, then $\liminf_{x\to\infty}f(x)\leq \liminf_{x\to\infty} g(x)$