Limits superior and inferior of measurable sets are measurable

Let \((X,\mcm)\) be a measurable space and let \(A_n\in\mcm\) for \(n\geq1\). Then \(\limsup_n A_n\) and \(\liminf_n A_n\), as defined by \(\bigcap_{N\geq1}\bigcup_{n\geq N}A_n\) and \(\bigcup_{N\geq1}\bigcap_{n\geq N}A_n\), belong to \(\mcm\).