I'd like to show that the Haussdorf-Dimension of the Cantor set is log(2)/log(3).

Although I have some Ideas how to do this, I'm not sure if any of it is correct. A little guidance would be greatly appreciated. I hope you guys are familiar with the terms I use. Otherwise I refer to Hausdorff measure - Wikipedia, the free encyclopedia

My idea was to make an upperbound for $\displaystyle \mathcal{H}^s_{\epsilon}(C_n)$

Starting with $\displaystyle \mathcal{H}_{\epsilon}^s(C_n)\leq (2/3)^n*\epsilon^{s-1}$.

Here $\displaystyle (2/3)^n$ is the total length of $\displaystyle C_n$ and $\displaystyle \epsilon$ the maximum diameter of a Cover-element of $\displaystyle C_n$.

Now we need to find out for which $\displaystyle s$ this upperbound exists, if we let $\displaystyle n\to \infty$ and $\displaystyle \epsilon\to 0$.

How to do this, I'm not sure yet.

Can anyone help me in the right direction?