Suppose $(X,\Delta)$ is a log canonical pair and $A$ is an ample divisor on $X$. Could you give me an esay proof of the fact that there exists $A'$ that is $\mathbb{Q}$linearly equivalent to $A$ such that $(X,\Delta+A')$ is again log canonical? Thanks a lot

If $B$ is a general member of a basepointfree linear system (say $mA$ for $m\gg 0$), then a log resolution $f:Y\to X$ of $(X,\Delta)$ is also a log resolution of $(X,\Delta+\frac 1m B)$ because $B$ will be transversal to all the strata related to the resolution. It follows that $f^*B=f^{1}_*B$ so the discrepancies don't change either and hence what has been (log) terminal/canonical, remains that. 

