The function $g:(0,\mathrm{\infty})\to \mathbb{R}$, is continuous, $g(1)>0$ and

$\underset{x\to \mathrm{\infty}}{lim}g(x)=0$

It is a fact that for every $y$ between 0 and $g(1)$ the function takes on a value in $(\text{}y,\text{}g(1)\text{})$

How would one show that if:

$\underset{e\to 0+}{lim}sup\{g(x):0<x<e\}=0$

then $g$ attains a maximum value on 0 to infinity. Does $g$ necessarily have to be bounded below for this to hold? Why?

$\underset{x\to \mathrm{\infty}}{lim}g(x)=0$

It is a fact that for every $y$ between 0 and $g(1)$ the function takes on a value in $(\text{}y,\text{}g(1)\text{})$

How would one show that if:

$\underset{e\to 0+}{lim}sup\{g(x):0<x<e\}=0$

then $g$ attains a maximum value on 0 to infinity. Does $g$ necessarily have to be bounded below for this to hold? Why?