High probability of $\sqrt{\log n}$ degree in random graph

Question

I’m stuck on a problem I’ve found in a book on random graphs. Appreciate any help:

Let $G_{n,p}$ be a random graph with $p=\frac{c}{n}$ for some positive $c$. Show that with high probability there is a vertex with degree at least $\sqrt{\log n}$.

Answer 1

A much stronger result is shown in this paper:

Riordan, Oliver; Selby, Alex, The maximum degree of a random graph, Comb. Probab. Comput. 9, No. 6, 549-572 (2000). ZBL0971.05101.