I am trying to prove that a $\displaystyle 2^{n} \times 2^{n} $ board with one square removed can be covered by L-shaped tiles(3 square L-shaped tile). Here is my proof:

20.Proof: We use induction on $\displaystyle n $. Base case: For $\displaystyle n = 1$, a $\displaystyle 2 \times 2$ square grid with any one square removed can be covered by exactly one L-shaped tile. Induction step: Suppose now as induction hypothesis that for some positive integer $\displaystyle k $, a $\displaystyle 2^{k} \times 2^{k}$ square grid with any one square removed can be covered using $\displaystyle r $ L-shaped tiles, were $\displaystyle r $ is some positive integer. Then a $\displaystyle 2^{k+1} \times2^{k+1}$ board with one square removed can be covered by $\displaystyle 2r $ L-shaped tiles. Conclusion: Hence, by induction, for a positive integer $\displaystyle n $, a $\displaystyle 2^{n} \times 2^{n} $ square grid with any one square removed can be covered by $\displaystyle 2r $ L-shaped tiles. $\displaystyle \square $

Is this proof correct?

Thanks