I am currently reading a proof in Pesaran's 2006 paper (Lemma A2 Eq:A11). There is a quantity $f(N,T):=\frac{F^T\bar{U}_w}{T}$ and the aim is to show that \begin{align} f(N,T) = O_p\left(\frac{1}{\sqrt{NT}}\right). \end{align}
They show that $Var(f(N,T)) = O(\frac{1}{NT})$ which would imply $f(N,T)=O_p\left(\frac{1}{\sqrt{NT}}\right)$ by Markov. Why dont they stop here and conclude the assertion? They rather argue as follows. 1. $ f(N,T) = O_p(\frac{1}{\sqrt{N}})$ , for fixed T and 2. $ f(N,T) = O_p(\frac{1}{\sqrt{T}})$, for fixed N. Then they conclude that $\sqrt{NT} f(N,T) \to O_p(1)$ in distribution which finishes their proof.
Thanks for your expertise in advance.