Corollary 2

Let $f$ and $g$ be two real-valued functions defined on a filtered set $X$. If $\lim f = +\infty$ and $\lim \inf g > -\infty$ and $f + g$ is defined, then $\lim (f + g) = +\infty$. If $\lim f = +\infty$ and $\lim \inf g > 0$ and $fg$ is defined, then $\lim fg = +\infty$.