Proof involving asymptotic complexity
The notations $a(n) = O(b(n))$ and $O(a(n)) = O(b(n))$ have different meanings:


$a(n) = O(b(n))$ means that $a(n)$ belongs to the collection of functions $O(b(n))$. That is, $a(n) \in O(b(n))$.
$O(a(n)) = O(b(n))$ means (in this particular case) that the two collections of functions $O(a(n))$ and $O(b(n))$ are the same. That is, $c(n) = O(a(n))$ iff $c(n) = O(b(n))$.  (Usually $O(a(n)) = O(b(n))$ actually means $O(a(n)) \subseteq O(b(n))$.)


Big O notation is thus ambiguous, but in practice there is usually no confusion.