Does this pattern continue $\lfloor\sqrt{44}\rfloor=6, \lfloor\sqrt{4444}\rfloor=66,\dots$? By observing the following I have a feeling that the pattern continues.
$$\lfloor \sqrt{44} \rfloor=6$$
$$\lfloor \sqrt{4444} \rfloor=66$$
$$\lfloor \sqrt{444444} \rfloor=666$$
$$\lfloor \sqrt{44444444} \rfloor=6666$$
But I'm unable to prove it. Your help will be appreciated.

Hint : 
We have $$\left(\frac{6\cdot (10^n-1)}{9}\right)^2=\frac{4\cdot (10^{2n}-1)}{9}-\frac{8\cdot (10^n-1)}{9}$$
Try to find out why this proves that the pattern continues forever.
