method of moments estimator and confidence interval - Geometric Distribution

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Question : enter image description here

i got to the point(its correct) that : $\hat{\theta}_{mm}$ = $\frac{1}{\bar{X}}$

$Var(\hat{\theta}_{mm})$ = $\frac{\bar{X}-1}{\bar{X}^3}$

now I need to construct the Confidence Interval for a given $\alpha$. i get that I need to use Asymptotic Normality but I don't get why in the solutions they divide by $\sqrt{n}$ : enter image description here enter image description here