 Note that \((\mathbf{K}^m_N(E))_{m\ge 1}\) is a decreasing sequence of events, in the sense that \(\mathbf{K}^{m+1}_N (E) \subseteq \mathbf{K}^m_N(E)\), for all \(m \ge 1\). It is also easy to check that if everyone knows \(E\), then \(E\) must be true, that is, \(\mathbf{K}^1_N (E) \subseteq E\). If \(\Omega\) is assumed to be finite, then if \(E\) is common knowledge at \(\omega\), this implies that there must be a finite \(m\) such that