I needed a result about the of a product of two sequences that I could not find in the references I had immediately at hand. Noting it down for future reference.
Let and be two sequences such that and .
Since it must be the case for all sufficiently large that . Since all the properties asserted in the theorem depend only on the tail behaviour of the sequences there is no harm in assuming that this inequality holds for all .
We make use of the definition of as the infimum of subsequential limits (Rudin, Principles of Mathematical Analysis, 3rd edition, Definition 3.16).
The theorem will be proved if we can show that any subsequence converges to a limit if and only if the corresponding subsequence converges to . If we can establish this fact, then, since , the infimum of the subsequential limits of will be times the infimum of the subsequential limits of .
When is finite, the required fact is a direct consequence of the usual theorems on limits of products and quotients.
For infinite we use the fact that has positive lower and upper bounds, so will be unbounded above or below if and only if is unbounded in the same way.
The condition is essential in the above theorem. Consider the following example: . . Then , , but .
We can get by with if we are willing to assume that is bounded. But not having to check boundedness when is a big help.