If and exists, we have . But if , do we have analog equation(I guess )? and what change should be made to conditions to achieve the analog equation? Formal source of reference such as textbooks or webpages is recommended. Thanks!
If and exists, we have . But if , do we have analog equation(I guess )? and what change should be made to conditions to achieve the analog equation? Formal source of reference such as textbooks or webpages is recommended. Thanks!
I would agree with you assertion, namely:
If and that . To do this we note that in general where is the set of all convergent subsequential limits of and .
So, let denote the set of all subsequential limits of and the multiplication by of each subsequential limit of . We claim that .
We may assume (since otherwise this is easy) that . So, let then for some injection . Thus, we note that (notice that if a limit converges to a values so does every subsequential limit) and thus by the assumption that both of the limits converge we see that and thus is times a subsequential limit of and thus .
Conversely, if then and once again since we assumed both of these limits exists we see that and thus
Thus,
NOTE: It's late and I haven't checked over the above too carefully. I have been known to make light night stupid errors.
Great! I checked it and found no problem. It's a rigorous and beautiful proof. From the proof, the conclusion holds even without any assumption regarding the sign of real terms and , only and finite is sufficient. We can also obtain similarly.
Thank you very very much, Drexel28! I would like to give you double thanks. Have a good dream:-)