Showing that the first order statistic of pdf $f(x)=2(x-\mu)$ is consistent for $\mu$

Question

I am given that $X_i$ are iid for $i=1,\cdots,n$ from pdf $f(x)=2(x-\mu)$ over $(\mu,\mu+1)$, and $0$ elsewhere. I am asked to show that the first order statistic is a consistent estimator for $\mu$.

So far, I have found the pdf of the first order statistic to be given by $$f_1(x)=2n(x-\mu)(1-(x-\mu)^2)^{n-1}$$ After that, I tried to show that $\mathbb E[(X_{(1)}-\mu)^2] \xrightarrow p 0$, since I understand that showing this demonstrates that $X_{(1)}$ is a consistent estimator for $\mu$. However, as part of trying to show this, I try to find $$\mathbb E[X_{(1)}^2] = \int_{\mu}^{\mu+1} t^22n(t-\mu)(1-(t-\mu)^2)^{n-1}dt$$ and I am unable to evaluate this integral.

Am I on the right track? Is there some clearly better way I should be going about this? If not, how do I proceed?

Answer 1

Following the suggestions in the comments, I reasoned as follows.

Let $1>\epsilon>0$. \begin{align*} P(|X_{(1)}-\mu|>\epsilon) &= P(X_{(1)}-\mu>\epsilon\\ &=\int_{\epsilon}^1 2nt(1-t^2)^{n-1}dt\\ &<\int_\epsilon^1 2n(1-\epsilon^2)^{n-1}dt\\ &<2n(1-\epsilon^2)^{n-1}\\ &\xrightarrow{n\to\infty}0 \end{align*}