Uniform random number generation from random bits

Question

I need to generate random number in a given interval using random bits. For example, If I want to generate a random number between 1 to 6, inclusive, I can concatenate three random bits which gives me a possible range of 000 to 111 (0 to 7), inclusive. My intuition says after concatenating 3 random bits, if the result falls between 000 and 101, inclusive, I can just return that number plus 1 (which essentially gives me a number between 1 and 6, inclusive). If I get 110 or 111 (6 or 7) after concatenating 3 random bits, I can retry again until I the number generated is in my desired range. How can I prove (or disprove) that the number I returned is indeed random?