Let X 1 , X 2 ,... be i.i.d. random variables with a distribution F. Show that X 1 ,...,X n are distinct with probability one for any n ≥ 2 if and only if F is continuous. (This statement is also true...

Let X₁, X₂,... be i.i.d. random variables with a distribution F. Show that X₁,...,X_n
are distinct with probability one for any n ≥ 2 if and only if F is continuous. (This statement is also true if these are random elements in a space S and “continuous F” is replaced by F({x}) = 0, x ∈ S). Hint: Use induction and