The Hausdorff d-measure of a set is non-zero and finite for at most one value of d and if so that d is the dimension. So to prove the Hausdorff dimension of the unit interval is 1 it is only necessary to show the 1-measure is finite and non-zero. So really it would be more use to learn how to prove my first sentence in the general case (I must admit I'd need to refer to a textbook for the proof myself), as then if you can have a lucky guess at the right answer for d, and the d-measure happens to be non-zero and finite you don't have to worry about any other possible dimensions.
Sorry if I am stating the obvious, as I probably am.