Let $f:(0,\infty)\rightarrow \mathbb{R}$ be a function verfying the following:
1) $f$ is stricly increasing.
2) For each positive sequence $(a_n)$, we have
$\lim_{n\rightarrow\infty}{a_n}=0$ iff $\lim_{n\rightarrow\infty}{f(a_n)}=-\infty$.
3) There exists $\theta\in (0,1)$ such that $\lim_{a\rightarrow 0^+}a^\theta f(a) =0 $
Let $\tau>0$, $\{x_n\}\subset (0,\infty)$ and $\{y_n\}\subset (0,\infty)$
such that $\{x_n\} ,\{y_n\}$ converge to $x,y>0$ and $f(x_n)\leq f(y_n)-\tau$. By using the upper limit, i want to show that $f(x)\leq f(y)-\tau.$
Thank you.
It is not correct.
Let $f$ be function satisfying condition 1,2,3 on interval $(0 , 1] $, It is easy to construct one. Now define $f$ in such way be a line with positive slope on $[1,2].$ Now continue define $f$ strictly increasing on $(2 , \infty )$ such that $f(z) \geq f(2) + 1 $, for all $z \in (2 , \infty )$. Then you have a jump at $2$.
Now by setting $x=y=2$ you have $f(x) = f(y) = f(2)$. Then easily make those sequences.