Understanding an estimate

Question

I am trying to understand an estimate from the post: Lower bound for expectation of maximum absolute value of standard normal random variables. In particular, in the answer posted there, we used the following fact: $$ \left( 1 - \frac{c}{n^{u^2}} \right)^n \leq \alpha^{1/u^2} $$ for some $\alpha \in (0, 1)$ only depend on $c$ and not $n$ or $u$. Here $c$ is an absolute constant that is potentially greater than 1, $u$ may be assumed to be large and $n \geq 2$. However, I have failed to see why this is true. Perhaps we can rewrite this as $e$ to the power of something, but I do not see how.