I am trying to understand a proof by David E Speyer on $\frac{logn}{n}$ being the threshold for connectivity of a random Graph $G \in G(n,p)$.
https://mathoverflow.net/questions/60075/connectivity-of-the-erdős-rényi-random-graph, it's the second answer.
In this proof, $p(n)=\frac{Clogn}{n}$ with $C>1$ is used to show that $$\sum\limits_{k=1}^{n/2}\binom{n}{k}(1-p)^{kn-k^2} \rightarrow 0. $$
Firstly, with $C>2$ this proof is quite easy.
Then the author completes the proof by showing it for $1 < C <2$ and there I get stuck. He chooses an $a \in \mathbb{R}$ such that $1-C(1-a)<0$.
From then, only $k<an$ is needed to be considered according to the author.
I don't understand why.
What I'm trying: $$\sum\limits_{k=1}^{n/2}\binom{n}{k}(1-p)^{kn-k^2}=\sum\limits_{k=1}^{an-1}\binom{n}{k}(1-p)^{kn-k^2}+\sum\limits_{k=an}^{n/2}\binom{n}{k}(1-p)^{kn-k^2} $$
and then maybe $$\sum\limits_{k=1}^{an-1}\binom{n}{k}(1-p)^{kn-k^2} \rightarrow 0?$$
Thanks for any help!