Let \(\left(I_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a continuous adapted process with values in \([0, \infty)\) and a.s. increasing sample paths. Set for \(u \geqslant 0\)

\[\sigma_{u}(\omega):=\inf \left\{t \geqslant 0: I_{t}(\omega)>uight\} \quad \text { and } \quad \tau_{u}(\omega):=\inf \left\{t \geqslant 0: I_{t}(\omega) \geqslant uight\}\]

Show that

a) \(\sigma_{u} \geqslant t \Longleftrightarrow I_{t} \leqslant u\) and \(\tau_{u}>t \Longleftrightarrow I_{t}

b) \(\tau_{u}\) is an \(\mathscr{F}_{t}\) stopping time and \(\sigma_{u}\) is an \(\mathscr{F}_{t+}\) stopping time.