Real analysis

Let $ \dot{\mathcal P } := \{ (I_i, t_i) \}_{i=1}^n $ be a tagged partition of $[a,b] $ and let $c_1 < c_2 $.

(a) If $u$ belongs to a subinterval $I_i$ whose tag satisfies $c_1 \leq t_i \leq c_2$ , show that $c_1 - || \dot{\mathcal P} || \leq u \leq c_1 + || \dot{\mathcal P} ||$ .

(b) If $v\in [a,b]$ and satisfies $c_1 + || \dot{\mathcal P} || \leq v \leq c_2 - || \dot{\mathcal P} ||$ , then the tag $t_i$ of any subinterval $I_i$ that contains $v$ satisfies $t_i \in [c_1, c_2 ]$ .

#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}`	$\boxed{123}$