Assume that f(n) and g(n) are asymptotically positive. Which of the following is correct?

If $f=o(g)$ and $g=o(h)$, then $\frac fg\to 0$ and $\frac gh\to 0$, so $\frac fh=\frac fg\cdot\frac gh\to 0$, which means $f=o(h)$; little-o is transitive. Two big-O bounds combine only to another big-O bound, so they cannot give $f=\omega(h)$, and two $\Omega$ lower bounds give a lower bound, not an upper bound. Concluding $f=\Omega(h)$ from two little-o relations contradicts $f=o(h)$.