True or false: if $f'(x)<0$, for all real numbers $x$, then $f$ decreases without bound.

Question

Determine whether the following statement is true or false. If $f'(x)<0$, for all real numbers $x$, then $f$ decreases without bound.

Answer 1

False. Consider $f(x)=e^{-x}$. Observe that $f'(x)=-e^{-x}<0$ for all $x\in \mathbb{R}$. Yet since: $$ \lim_{x\to\infty} e^{-x}=0 $$ we know that $f$ is decreasing but is bounded below by $0$.

Answer 2

Hint: Consider this plot of $f(x)=e^{-x}$: