Product of Big-O Estimates/Sequences

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Theorem

Let $(a_n),(b_n),(c_n),(d_n)$ be sequences of real or complex numbers.

Let:

$a_n = O(b_n)$
$c_n = O(d_n)$

where $O$ denotes big-O notation.


Then $a_nc_n = O(b_nd_n)$.


Proof