If you need to "prove" without reference to the documentation of how floating point values are stored, consider empirical proof:
R = 1 + rand(1,10000000); %say 10 million attempts;
D = min(diff(unique(R))); %sort the tries. Find the adjacent differences. What is the smallest difference?
Any particular answer you get with this would be an upper-bound on the difference between adjacent values in the interval.
Now, how do you know that there are not locations in the interval where the difference might be smaller? You do not, with certainty -- but you can generate more random numbers. You can also use a minor variation of The Birthday Paradox in order to calculate the confidence value that you have found the minimum.
So then, how do you know that there are no locations where the difference between adjacent values is not more than the calculated value? Well, if you take
R = rand(1,1000000);
U = unique((R+D) - R);
then what information would U give you?
