Let be vectors of – dimensional Euclidean space such that . Prove that there exists a permutation of the integers such that
for each .
Let be a positive and strictly decreasing sequence such that . Prove that the series
We shall begin with a lemma.
Lemma: Let . It holds that
Proof: We are using induction on . For it is trivial. Suppose that it holds for . Then
Thus it holds for and the lemma is proved. Since and I can find such that for all . But then
where in the first inequality the lemma was used.
The exercise can also be found at mathematica.gr . It can also be found in Problems in Mathematical Analysis v1 W.J.Kaczor M.T.Nowak as exercise 3.2.43 page 80 .
(a) Let . Evaluate the integral:
(b) Let and let be a sequence defined as
Find the limit of .
(a) We have successively:
where denotes the lower incomplete Gamma function.
(b) We are invoking Tonelli’s theorem. Hence the limit we seek is equal to
For any positive integer , let denote the closest integer to . Evaluate:
We begin by the simple observations that
. Combining this we get that