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What is the growth rate in n of the denominator of the rational $r_n$, $n=1,2,3,\ldots$ in $1>r_n>1-e^{-n}$ with the smallest denominator?

If $I_n$ is a (sufficiently random) sequence of disjoint intervals in R>0, whose lenght (ie elementary measure) is given by a f(n) decreasing to 0, with $\sum_n f(n) = \infty$, what is the expected growth rate of the lowest denominator of all rationals in $I_n$ ?

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    For the first question, isn't the answer trivially $\approx e^n$?2011-12-23
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    I ask question, dont respond with question please.2011-12-23
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    I don't know if it was intended that way, but the tone of the response to Srivatsan is very hostile.2011-12-23

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