Consider these expressions. $B_{2}(x)=x\ln(2)+ O(\ln(x))$ and $\psi(x)\geq B_{2}(x)$
In the next step, $ \psi(x)\geq x\ln(2)+ O(\ln(x)) ............(1)$
What does the expression (1) actually mean, mathematically? From what I know, when a big oh notation is involved, there is an equal to sign between LHS and RHS of the expression and the definition of big oh notation makes sense, but I am not to understand what it means when an inequality in involved like in (1).
$O(f)$ is set of functions with well known definition(considering non-negative case): $h \in O(f)$ iif $\exists C>0, \exists$ neighbourhood for $x$, such that $h(x) \leqslant C f(x)$
So, equality means belonging: $B_{2}(x)=x\ln(2)+ O(\ln(x))$ is same with $B_{2}(x)\in x\ln(2)+ O(\ln(x))$ where in right side is again set $g+O(f)=\{g+\phi \colon \phi \in O(f) \}$.
Now, $\psi \geqslant g+O(f)$ we can understand as $\psi \geqslant h$ for $\forall h \in g+O(f)$.