Let $(q_n)_{n>0}$ be a real sequence such that $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?