Hello! Hope all is well. This is a probability theory class(masters level). I need a correct detailed answer for this question please!

1. Let (Xn) be a sequence of random variables on a probability space (S, F, P). Define a function X : 0 - R by X(w) = limn > Xn(w). We assume that this limit exists for each w E . Prove that X is a random variable. Conclude that if (Xn) is a sequence of random variable such that for each w E n the infinite sum En- Xn(w) converges, then X(w) = _. Xn(w) is a random variable