[Undergrad] Proving that certain geometric random variables are independent.

Suppose that you are told to toss a die until you have observed each of the six faces. What is the expected number of tosses required to complete your assignment? [Hint:If Y is the number of trials to complete the assignment, Y = Y_1 +Y_2 +Y_3 +Y_4 +Y_5 +Y_6 , where Y_1 is the trial on which the first face is tossed,Y_1 = 1, Y_2 is the number of additional tosses required to get a face different than the first, Y_3 is the number of additional tosses required to get a face different than the first two distinct faces, ...,Y_6 is the number of additional tosses to get the last remaining face after all other faces have been observed. Notice further that for i = 1,2,...,6, Y_i has a geometric distribution with success probability (7 − i)/6.]

This is from exercise 5.88 in Mathematical Statistics with Applications by Wackerly et al. (emphasis mine). I can solve the above problem easily enough, but there is a later problem which asks you to prove that Cov(Y_i, Y_j) = 0 for all i,j such that i is not equal to j. I figured you could do this by proving that they are independent and it seems intuitive to me that they are in fact independent (the number of additional tosses shouldn't depend on the number of tosses that came before or after), but I am having trouble showing this. I am not given a joint or conditional probability function for Y_i and Y_j so I am not sure how to prove independence or even calculate E[Y_i*Y_j]. Any thoughts or hints?