Can a bounded function be monotonically increasing for all x>0?

Question

Can we have a function such that it is always increasing for all x>0 and is also bounded above by some real number? How do we prove or disprove this statement?(The above statement is not a problem itself, this concept is required in another problem, so I want some help regarding it!)

Answer 1

Consider a function such as $f(x)=1-e^{-x}$, or $g(x)=\arctan{(x)}$, or $h(x)=1-\frac{1}{x}$. These all satisfy the requirement you've given, being bounded above by $1, \frac{\pi}{2}, 1$, respectively.