Why $O(\lambda_\text{max}^{\tau/2})=O(1/n^2)$ for $\tau=\frac{4\ln n}{1-\lambda_\text{max}}$?

Question

It appears in the proof of Lemma 9 in 'Bounds on the cover time' by A. Z. Broder and A. R. Karlin.

Answer 1

I am assuming that $0 < \lambda_{max} < 1$.

Hint : Prove that $\frac{2 ln(\lambda)} {(1-\lambda)} < -2$.