let $\displaystyle X_1,...,X_n$ are iid $\displaystyle U(0, \theta)$. I am trying to show that $\displaystyle \sqrt{T_n} \rightarrow^{p} \sqrt{\theta}$(converges in probability to), where $\displaystyle T_n$ is the $\displaystyle largest\;order\; statistic$

what I did was using the pdf of max order statistic find

$\displaystyle E(T_n)\;=\;\dfrac{n\theta}{n+1}$

$\displaystyle E({T_n}^2)=\dfrac{n\theta^2}{n+2}$

so, $\displaystyle E\{(T_{n}-\theta)^2\} = \dfrac{2{\theta}^2}{(n+1)(n+2)}$, which converges to 0, as n goes to infinity,

so we can conclude by weak law of large numbers that $\displaystyle T_n \rightarrow^{p} \theta \;as\; n \to \infty$

and this $\displaystyle \sqrt{T_n} \rightarrow^{p} \sqrt{\theta}$ follows from the theorem of continuous mapping....

is my approach to the proof correct?

thank you.