Let the sample space be $[0,1]$ with the Borel sigma algebra and the Lebesegue measure.
$$ X_n(w) = \begin{cases} 1, & \text{for $ 0\le w \le 1-\frac {1}{n}$ } \\ n, & \text{otherwise } \\ \end{cases} $$ It surely converges a.s. to 1. Does the following converges in $L^1$? I don't think so. It looks like to me that $lim \rightarrow E[|X_n-1|]=1$. Am I correct?
This is indeed right. The random variable $\left\lvert X_n-1\right\rvert$ takes the value $0$ with probability $1-1/n$, and the value $n-1$ with probability $1/n$ hence $\mathbb E\left[\left\lvert X_n-1\right\rvert\right]=(n-1)/n$, which converges to $1$, as said in the opening post. Maybe add a sentence, that if there is a convergence in $L^1$, then it is necessarily to the almost sure limit.