I hope anyone can explain this to me, who is familiar with the Hausdorff measure:

See:

Hausdorff measure - Wikipedia, the free encyclopedia
Wikipedia says:$\displaystyle \mathcal{H}_{\delta}^d(S) $ is monotone decreasing in $\displaystyle \delta$, and I agree. Since the larger $\displaystyle \delta$, the more covers of U are permitted...

Then they conclude: Hence $\displaystyle \lim_{\delta\to 0}\mathcal{H}_{\delta}^d(S)$ exists. Why? Doesn't it only imply that $\displaystyle \lim_{\delta\to\infty}\mathcal{H}_{\delta}^d(S)$ exists?

I don't understand this at all.