Showing that there do not exist uncountably many independent, non-constant random variables on $ ([0,1],\mathcal{B},\lambda) $.

I have this problem in my assignment:

Show that there do not exist uncountably many independent, non-constant random variables on $ ([0,1],\mathcal{B},\lambda) $, where $ \lambda $ is the Lebesgue measure on the Borel $ \sigma $-algebra $ \mathcal{B} $ of $ [0,1] $.

Can someone please help me to solve this?

Assume that $(X_i)_{i \in I} $ are independent random variables. Let $Y_i := X_i \cdot 1_{|X_i|\leq C_i} $, where $C_i>0$ is chosen so large that $Y_i \not\equiv c_i $ for some constant $c_i $. This is possible since the $X_i $ are non-constant.

Then the $(Y_i - \Bbb{E}(Y_i))_i $ form a family of independent and hence orthogonal random variables. Note that the $Y_i $ are bounded and thus contained in $L^2$.

But the separable (!) space $L^2 ([0,1],\lambda) $ can only contain countably many elements which are mutually orthogonal. Hence, $I $ is countable.