Let be a real sequence such that , , and . Prove that
Lemma: Let be a bounded sequence of, say, complex numbers and let be another complex number.
where the latter means “convergence on a set of density ” i.e. there exists a subsequence such that and is dense i.e.
Proof: The proof is omitted because it is too technical.
All we need is that the logarithm is continuous and that as well as are bounded sequences.
Note: We can simplify the conditions to and .
The exercise along with the solution may be found on AoPS.com . The proof of the claim may also be found there.