Answer the following quesiton, thank you

Problem 12 (Problem 1.14 from the book). Let X, Y, X1, X2, ... be real-valued random variables on an event space (2, F). Prove the following statements: (a) (X, Y) : 0 - R', (X, Y) (w) := (X(w), Y(w)) for w E 2, is a random variable. (b) X + Y and XY are random variables. (c) super Xn and lim sup, 20 Xn are random variables (taking values in R). (d) {X = Y} ( F, {limn woo Xn exists} ( F, {X = limn +0 Xn} EF