Let A1,A2,A3,… be a sequence of sequences where each
Ai=ai,1,ai,2,ai,3,…
Each sequence Ai converges and in particular as t→∞, ai,t→Li for every i. We also have that in the limit as i→∞ the limits of these sequences converge to 1. I.e. as i→∞, Li→1.
QUESTION: When does the following occur?
In the limit as t→∞
∞∏i=1ai,t→∞∏i=1Li
I know how to prove that the limit of the product of two (convergent) sequences is the product of the limit of those sequences. I cannot see whether this argument extends to infinite products. Also I am not sure how to use the fact that the Li converge to 1.
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