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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
1 year, 2 months ago