Friday, 15 April 2016

real analysis - Is it true that prodinftyi=1ai,trightarrowprodinftyi=1Li when ai,trightarrowLi for every i, and Lito1?

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,tLi for every i. We also have that in the limit as i the limits of these sequences converge to 1. I.e. as i, Li1.



QUESTION: When does the following occur?



In the limit as t
i=1ai,ti=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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