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

#Calculus

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$