Asymptotically confidence interval of level $1-2\alpha$

For fixed $k$, the variables $(2k+1)\hat{f}_k(\lambda)$ are asymptotically distributed according the Gamma distribution with shape parameter $2k+1$ and mean $(2k+1)f_X(\lambda)$. It turns out that this suggests a confidence interval of the form \begin{align} \bigg(\frac{(4k+2)\hat{f}_k(\lambda)}{\chi^2_{4k+2,1-\alpha}}, \frac{(4k+2)\hat{f}_k(\lambda)}{\chi^2_{4k+2,\alpha}} \bigg), \qquad (*) \end{align} where $\chi^2_{k,\alpha}$ the $\alpha$-quantile of the chi-square distribution with $k$ degrees of freedom.

I want to show that this interval is asymptotically of level $1-2\alpha$.

I know that a $\Gamma(m/2,1/2)$-distribution is identical to a $\chi^2_m$-dsitribution. So in $(*)$ we have a fraction of two object which are Gamma-distributed. However, I do not see how to make ends meet?