Independence of two disjoint sums of independent Random variables

Suppose in a probability space $$(X,\Omega,\mu)$$, let $$X_1,X_2,...X_n$$ be independent random variables. How do I prove that $$X_1+X_2+...+X_{i-1}$$ is independent of $$X_i+...+X_n$$ for some $$1\leq i \leq n$$ ?

• Write down explicitly the definition of "independence of those two disjoint sums" and see if you can verify it. It may also help to write down explicitly the definition of "$X_1, \ldots, X_n$ are independent." – angryavian Nov 1 '18 at 3:28
• math.stackexchange.com/questions/8742/… – d.k.o. Nov 1 '18 at 3:55

Demonstrate that $$X_a+X_b, X_c$$ are independent if $$X_a,X_b,X_c$$ are mutually independent.
• I got your point. But how can I show that $X_1+X_2$ independent of $X_3$ here ? – Ganesh Gani Nov 1 '18 at 9:29