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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 \(t_i \in [c_1, c_2 ] \).

Note by Syed Subhan Siraj
7 months, 1 week ago

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