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I'm working currently with a Poisson process trying to proove Renyi's Theorem, so far I want to show that $\prod_{i=1}^{k_n}[z + (1-z)e^{-\mu(A_{n_i})}] \to e^{-(1-z)\mu(A)}$ as $\mu(A_{n_i}) \to 0$, where $A = \bigcup_i^{k_n} A_{n_i}$, the $A_{n_i}$ are disjoint and $\mu$ is a Radón measure.

So first I have to show that $[z + (1-z)e^{-\mu(A_{n_i})}] \to e^{-(1-z)\mu(A_{n_i})}$ using that for small $\delta$ $1-\delta<e^{-\delta}<1-\delta + \delta^2$, I already have the lower bound, but I'm having trouble with getting the upper bound.

I've already tried other forms of the Taylor series of $e^{-x}$ but it doesn't seem to work. The suggestion of bounding the exponential like above is in "An Introduction to the Theory of Point Processes. Volume I: Elementary Theory and Methods"-Daley, D.J. and Vere-Jones,D., p.32.

Edit1: I was looking at it wrong, instead of $\prod_{i=1}^{k_n}[z + (1-z)e^{-\mu(A_{n_i})}] \to e^{-(1-z)\mu(A)}$ it is $\prod_{i=1}^{k_n}[z + (1-z)e^{-\mu(A_{n_i})}] \approx e^{-(1-z)\mu(A)}$, which is easy to solve taking the quotient.

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