real analysis - Is it true that $prod_{i=1}^{infty} a_{i,t} rightarrow prod_{i=1}^{infty}L_i$ when $a_{i,t} rightarrow L_i$ for every $i$, and $L_ito 1$?

Let $A_1,A_2,A_3,\dots$ be a sequence of sequences where each
$$A_i = a_{i,1},a_{i,2},a_{i,3},\dots$$

Each sequence $A_i$ converges and in particular as $t \rightarrow \infty$, $a_{i,t} \rightarrow L_i$ for every $i$. We also have that in the limit as $i \rightarrow \infty$ the limits of these sequences converge to 1. I.e. as $i \rightarrow \infty$, $L_i \rightarrow 1$.

QUESTION: When does the following occur?

In the limit as $t \rightarrow \infty$
$$\prod_{i=1}^{\infty} a_{i,t} \rightarrow \prod_{i=1}^{\infty}L_i$$

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 $L_i$ converge to 1.