Dominated convergence theorem

Hi

i am trying to solve a past exam paper question for an exam i have next week. Ive spent a while trying to do it but i dont think its right. The problem with this question is that most similar problems i have solved dont have X(w) explicitly in the question

Xn(w) = X(w) if |X(w)|> n or 0 if |X(w)|>= n

Show that Xn(w) tend to 0 as n tends to infinty

Deduce that

lim(as n tends to infinity) Integral of X(w) dP(w) = 0 wth integration limits {w:|X(w)|>n}

( sorry but dnt know how to use maths edit on these forums)

My answer

Suppose there exists an N>0 st n> or = N ==> n > |X(w)| for any fixed w £ [0,1]. ( my reasoning is that |X(w) is just a number so for nu sufficiently large this will be true)

So limXn(w) is zero for all fixed w.

So now i can use dominated convergence thm, with |Xn(w)|< or = |X(w)|

so limE[Xn(w)] = E[X(w)]

but from last part limXn(w) = 0 as n tends to infinty

This is where i get stuck

please help