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The binomial coefficient in the RHS enumerates the subsets of size of . The LHS does the same thing, but choosing first the largest element of , then its second-to-largest element , until choosing its smallest element .
Let be the integral. Note that
For the integral we apply the substitution . Then, and
and similarly by applying the change of variables at the second integral we get that
Adding equations , we get that
and the result follows.
Let . Prove that:
We’re applying the change of variables and thus,
Let be a convergent series of positive terms. Prove that there exists a strictly increasing sequence which is also unbounded such that the series also converges.
We set and we observe that
That’s all folks.