Distribution of the decimals of a random number

[libelec]

Homework Statement

Let U = 0.X₁X₂X₃... be a random number in (0,1].

1) Find the distribution of every decimal digit X_i, i = 0,1,2...

2) Show that they are independent of each other

The Attempt at a Solution

I could use a hint for N°2. I have an idea, but I think it's wrong:

If X₁ and X₂ are independent, then P(X₂ = x2 (intersection) X₁ = x1) = P(X₂ = x2)*P(X₁ = x1).

Since P(X₂ = x2 (intersection) X₁ = x1) = P(0.x1x2... < U <= 0.x1(x2+1)...) = 0.x1(x2+1) - 0.x1x2 = 0.01 = 0.1*0.1 = P(X₂ = x2) * P(X₁ = x1) , that should prove it. Then, I could invoke the induction principle. But something sounds off.

Is this correct?

 

[libelec]

Anybody?