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Is this statement true?

For each two functions $f$ and $g$, where $f \in o(g)$, there exists a function $h$ where $f \in o(h)$ and $h \in o(g)$

Please note that I am using small $o$ notation.

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Yes.

For example, let $h=\sqrt{fg}$, assuming $f$ and $g$ are positive.

If not, we can let $h(x)=\max(\sqrt{|f(x)g(x)|}, g(x)/(|x|+1))$. The term, $g(x)/(|x|+1)$ makes sure that $h(x)$ will stay positive eventually.


Exercise. For each two functions $f$ and $g$, where $f \in \omega(g)$, there exists a function $h$ where $f \in \omega(h)$ and $h \in \omega(g)$.

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