Given $X_1,\ldots, X_n$ are i.i.d rvs with pdf $f_{X}(x) = 3(1-x)^2I_{(0,1)}(x)$. Define $A_n = n^{\frac{1}{3}} (1- max\left\{X_1,\ldots, X_n\right\})$ and $K_n = \sqrt{A_n}$. Find the limiting distribution of $K_1, K_2,\ldots$.
My attempt: I was able to show that $K_n\rightarrow K$ in distribution whose pdf is $f_{K}(t) = 3t^2e^{-t^3}I_{(0,\infty)}(t)$. Now, by defining the function $h(x) = \sqrt{x}$, which is continuous over $(0,\infty)$, then by a famous theorem, $\sqrt{K_n}\rightarrow \sqrt{K}$ in distribution. It's easy to see that $\lim_{n\rightarrow \infty} F_{\sqrt{K_n}}(t) = 1-e^{-t^6} $ for $\ t\in (0,\infty)$, $= 0$ otherwise. Thus, the limiting distribution of $ F_{\sqrt{K_n}}(t)$ follows a Weibull distribution with mean $\Gamma(7)$, whose pdf is $\ f_{\sqrt{K}}(t) = 6t^5e^{-t^6}I_{(0,\infty)}(t)$
My question: Is my solution above correct?