Is there any upper-bound function such as $\frac{k}{x}$ for monotonically decreasing functions?

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Could you please tell me if the following statement is true?

"Any monotonically decreasing function can be upper-bounded by function $f(x)=\frac{K}{x}$, i.e, if $g(x)$ is a monotonically decaying function then there is a positive real $K$ such that $g(x)≤f(x)$ for $x>0$."

If so, is there any proof for this statement or similar ones that can be found? Any references would be much appreciated. Thanks