The following is a simple consequence of the definition of Hausdorff measure: if and , then , and if and then . This reduces your present situation to showing that , which is a straightforward computation.

Edit:

Fix a and let be any covering of F so that for each j. Now .

So by definition of the Hausdorff measure, .

That is, .

Inffing over all such coverings, we're led to

.

By definition, . Take the limit as delta tends to 0 and conclude that since s < 2, the left hand side tends to infinity if the 2 dimensional Hausdorff measure of F is positive.