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Let $(q_n)_{n>0}$ be a real sequence such that $0 for all $n>0$ and $\lim_{n\to \infty} q_n = 0$.

For each $n > 0$, let $X_n$ be a random variable, such that $P[X_n =k]=q_n(1−q_n)^{k−1}, (k=1,2,...)$.

Prove that the limit distribution of

$\frac{X_n}{\mathbb{E}[X_n]}$

is exponential with parameter 1.

I see that $\mathbb{E}[X_n] = \frac{1}{q_n}$ but after that I don't really know where to go from there. Are there any tips please?

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