Proving the relationship between the degree and diameter of a graph and the Moore bound

Question

I am trying to show that the asymptotic Moore bound $n=O (\Delta ^D) $ for a graph of diameter $D $ and maximum degree $\Delta $ implies that $\Delta$ and $D $ cannot both be less than $$(1+o (1)) \frac {\log n}{\log\log n} $$ and I know the proof of this will likely be some sort of contradiction but I am struggling to properly deal with the constants involved. Any help would be appreciated.