I am trying to prove something, and it will help e greatly to show that the maximum degree of any vertex, $d$ is a connected simple graph, goes up asymptotically linearly along with the number of edges in the graph.
I came up with a somewhat hand-wavey "proof", and I wonder if it is anywhere near correct:
For fixed $|V|=n$, as $m$ grows, so does $d$, linearly, with constant average $\frac{1}{n}$, as by the pigeonhole principle, for every addition of n edges, the maximum degree has to increase by at least 1.
The other direction is easily $m \le \frac{dn}{2}$.
I wonder if this is at all correct, and if so, how to improve the proof