I think it's basically sound. You could make it shorter by only getting a contradiction at the left or right end of the interval, instead of both ends. But I think an easier way would be to say that any finite collection of real numbers can be well-ordered consistent with the standard order, so if you take two "adjacent" points a,b then the open interval (a,b) would have to be empty. One way or another, it comes down to the Archimedean property of the reals.