Is $2^{2n} = O(2^n)$?

Question

Is $2^{2n} = O(2^n)$?

My solution is:

$2^n 2^n \leq C_{1}2^n$

$2^n \leq C_{1}$,

TRUE.

Is this correct?

Answer 1

If $2^{2n}=O(2^n)$, then there is a constant $C$ and an integer $M$ such that for all $n\ge M$, the inequality $2^{2n}\le C 2^n$ holds.

This would imply that $2^n\cdot 2^n\le C 2^n$ for all $n\ge M$, which in turn implies $$\tag{1} 2^n\le C \quad {\bf for\ all } \quad n\ge M. $$ Can such $C$ and $M$ exist? Note the right hand side of $(1)$ is fixed, and the left hand side...

Answer 2

$x^n=o(y^n)$ iff $x<y$, as $(\frac{x}{y})^n\rightarrow 0$. Here 4 and 2, so no.