let $ (\xi_i)_{I \ge 1} $ be identical independent random variable, taking value in $ (1,3]$

can we show:

$P( \exists N \in \mathbb{N}, \text{ s.t. } \forall k \ge 0, \prod_{i=1}^{N}\xi_{i+kN} > N ) =1 ?$

i.e, almost surely, for each consecutive block with length $ N$, its product is greater than its length?

I think law of large number may give clue for this problem, but do not know how to prove it. Thanks!