I'm reading this proof (page 7), and get confused by the ending part , quoted here:
Take $c$ a constant less than $\sqrt{2}$. Then
Since $var(x) \leq n^{3-2c^{2}}+n^{2-c^{2}}, E(x) \leq n^{2-c^{2}}$, and $E^{2}(x) \leq n^{4-2c^{2}}$, it follows that $var(x) \leq E^{2}(x)$.
What's the reason it follows that $var(x) \leq E^{2}(x)$? Is it because $E^{2}$ is on the order of 4 while $var(x)$ is on the order of 3?
The conclusion cannot possibly be valid the way you have put it, because you have $$Var(x)\le\hbox{something}\ ,\quad E^2(x)\le\hbox{something}\ ,$$ and you are asked to conclude $$Var(x)\le E^2(x)\ .$$ This is in effect the same as asking "$a\le10$, $b\le20$, which is bigger, $a$ or $b$?" There is no way to know.
I suggest you look carefully at earlier parts of the paper, perhaps there is more information which will help.