I'm trying to prove a fiddly lemma for homework, but getting absolutely nowhere with it. Here, $G_{n,p}$ and $G_{n,m}$ represent, respectively, random graphs on $n$ vertices where the number of edges is binomially distributed with probability $p$ (i.e. any edge in the graph is present with probability $p$ and $N = {n \choose 2}$ possible edges, mean $Np$), and random graphs on $n$ vertices uniformly distributed across all possible such graphs with $m$ edges. We say $G_{n,p}$ has a property $Q = (Q_n)$ 'WHP' (with high probability) if $\mathbb{P}_{n,p}(Q_n) \to 1$ as $n \to \infty$ (similarly for $G_{n,m}$) - for example, this might be that the graph is connected, or is a forest.
Now, the lemma I want to prove is as follows:
Let $(Q_n)$ be a property of graphs. Then $G_{n,p}$ has $Q$ WHP iff for every $x > 0$ and $\epsilon > 0$ we have (as $n \to \infty$): $$\#\{m: |m-pN| \leq xn\sqrt{pq} \text{ and } \mathbb{P}_{n, m}(Q_n) < 1 - \epsilon\} < \epsilon n \sqrt{pq}$$
Now I've spent a very long time trying to chisel away at thie "lemma" but sadly I'm having no luck. In the $\Rightarrow$ direction, we have to translate from some condition in $G_{n,p}$ to one in $G_{n,m}$; the latter condition is, conceptually, on showing that the number of $m$ "close to the mean" which have a low $\mathbb{P}(Q_n)$ is small. One thought I had was to try and convert from our binomial distribution to its normal approximation and then use that (since of course $n\sqrt{pq}$ is a close approximation to the standard deviation), but again no progress.
It was suggested that it might help to observe that if $\omega (n) \to \infty$ as $n \to \infty$ then WHP,
$$|e(G_{n,p}) - pN| \leq \omega(n)n\sqrt{pq}\;,$$
but I couldn't complete the proof - nothing has worked so far. Could anyone help please? I've spent ages on this and not managed either direction, so the more detailed assistance you could provide the better. Many thanks for the help.