Let \(\dot{\mathcal P } \) be a tagged partition of \( [0,3] \).

(a) Show that the union \(U_1\) of all subintervals in \(\dot{\mathcal P} \) with tags in \([0,1]\) satisfies \( [ 0, 1- ||\dot{\mathcal P} || \subseteq U1 \subseteq [ 0, 1 + || \dot{\mathcal P} || ] \).

(b) Show that the union \(U_2\) of all subintervals in \(\dot{\mathcal P} \) with tags in \([1,2]\) satisfies \( [ 1 + ||\dot{\mathcal P} || , 2- ||\dot{\mathcal P} || \subseteq U2 \subseteq [ 1 - ||\dot{\mathcal P} ||, 2 + || \dot{\mathcal P} || ] \).

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