Define a sequence $\{X_n\}$ of random variables on $\Omega=[0,1]$ by
$X_n(\omega)=\begin{cases}n,&\omega\in[1-\frac{1}{n}, 1] \\0,& \omega\in[0,1-\frac{1}{n}) \end{cases}$
My question concerns whether the dominated convergence theorem is suitable for analysis of expectations of this sequence. As $n\to\infty$, the sequence approaches $0$ at all points in $\Omega$ except at the end point $1$, where the sequence diverges. So does this mean that we cannot use the DCT, since we cannot find a r.v. that bounds the sequence at all points in $\Omega$?