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I am referring this calculation https://stats.stackexchange.com/questions/284569/convergence-of-the-maximum-likelihood-estimator-of-a-uniform-parameter

Here the accepted answer says that $\left( \dfrac{\theta - \epsilon}{\theta} \right)^n$ tends to 0 as $n \to \infty$

I wonder which rule follows this. I looked into different formulas for limit in https://www.cuemath.com/limit-formula/, but could not figure this out.

Jean Marie
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Brian Smith
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  • If both $\theta, \epsilon$ are positive, then $0<\frac{\theta - \epsilon}{\theta} <1$. To be convinced, take your favorite number in $(0,1)$ and raise it to a high enough power, see what happens. Here is a more detailed proof https://math.stackexchange.com/questions/197522/how-to-prove-that-lim-n-to-infty-n-xn-0-when-0x1 – JBuck Feb 05 '23 at 15:17

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