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# Real analysis

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 $$[ 1 + ||\dot{\mathcal P} || , 2- ||\dot{\mathcal P} || \subseteq U2 \subseteq [ 1 - ||\dot{\mathcal P} ||, 2 + || \dot{\mathcal P} || ]$$.

Note by Syed Subhan Siraj
1 year, 7 months ago