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I would like to show that $2^{2n} \notin O(2^n)$. I would like to avoid using limits. My current approach is the following:

$$2^{2n} = 4^n \leq c \cdot 2^n \implies 2^n \leq c$$

Which is not possible, hence it is $2^{2n} \notin O(2^n)$. Is this valid?

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    $\begingroup$ Yes, it is valid. $\endgroup$
    – coffeemath
    Sep 14 '20 at 1:49

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