Show $0.35\left(\frac{2 a}{c}\right)^{a /(a-1)} \exp \left(-\frac{2}{a-1}\right) \frac{c}{2a} \to 0$, for $a(n)\geq 1.1, c(n)=$ $o(a n^{1-1 / a})$.

Question

I’ve just got back to study some probabilistic graph theory, so I’m not sure how to deal with proving the following lemma.

Lemma. Let $a=a(n), c=c(n)$ be functions of $n$ such that $a(n) \geq 1.1$ and $c(n)=$ $o(a n^{1-1 / a}).$ Then, for $p(n)=c(n) / n$, w.h.p. $G(n,p)$ contains no subgraphs with $s$ vertices, $s \leq 0.35\left(\frac{2 a}{c}\right)^{a /(a-1)} \exp \left(-\frac{2}{a-1}\right) n$, and more than $as$ edges.

Let $S$ be a subgraph of $G(n,p)$ on $s$ vertices. Then, using Markov’s inequality, I believe I will have to show that $$Pr(e(S) \geq as) \leq \frac{E(e(S))}{as} \approx \frac{sc}{2an} \to 0,$$ or $$Pr(e(S) \geq as) \leq 0.35\left(\frac{2 a}{c}\right)^{a /(a-1)} \exp \left(-\frac{2}{a-1}\right) \frac{c}{2a} \to 0.$$

What can I do with this expression?