Let $W$ be Gaussian Wigner matrix, and it is well-known that its operator norm satisfies
$$P(\|W\|\leq 2\sqrt{n}+t\geq 1-2\exp(-ct^2)).$$
My question is too basic but I cannot figure it out:
What is the operator norm of $f(n)W$, for example $\sqrt{n}W$?
From above inequality we have
$$P(\|\sqrt{n} W\|\leq 2\sqrt{n}\sqrt{n}+t)\geq 1-2\exp(-C(t/\sqrt{n})^2)$$
But then I don't know how to proceed. I think I need to do some transformation such that the right hand side becomes $ 1-2\exp(-ct^2))$.