$$ 3. Given a partition $P=\left\{a=x_{ \leq x_{1} \leq x_{2} \leq \ldots \leq x_{n-1} \leq x_{n}=b ight\}$ of $[a, b]$, we define its diameter by $$ \mu (P)=\max \left\{\left|x_{i}-x_{i=1} ight|: i=1,2, \ldots, n ight\} $$ This is a measure of how fine the partition is. Suppose $f:[a, b] ightarrow \mathbb {R}$ is bounded. Given a partition $P$, and evaluation points $t_{i} \in\left(x_{i-1), x_{i} ight]$, we define the Riemann Sum SCP, f)=\sum_{i=1}^{n} f\left(t_{i} ight) \Delta x_{i} $$ Prove that $f \in \mathcal[R) ([a, b] $ if and only if there exists $A Vin \mathbb {R}$ such that for each sequence $\left\{P_{n} ight Vin Vin \mathbb{N}$ of partitions such that $\mu\left(P_{n} ight) ightarrow 05, the sequence $\left\{\left(P_{n}, f ight) ight\}$ converges to $A$, for any choice of evaluation points in each $P_{n}$. Show that in that case $$ A=\int_{a}^{b} f(x) dx $$ SP.SD.3021