1
$\begingroup$

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 $\liminf_{x\to\infty}f(x)\leq \liminf_{x\to\infty} g(x)$

  • 2
    Write down the definition of $\liminf$. Then try to assume that what you wrote is false. That is there is a sequence of $x_n$ going to infinity with $f(x_n)<=M$.2012-04-02
  • 0
    I know that the first argument is true for sequences: If $\liminf_{n\to \infty}x_{n}>M$ then there exists $n_{o}$ such that $x_{n}>M$ for all $n\geq n_{o}$. But is it true also for functions!?2012-04-02
  • 0
    Using Alex R.'s argument, a contradiction will happen. However, I'm going to do this question in another way for fun.2016-02-11

1 Answers 1