- Integrate $f(x)=x$ for $x\in [0,1]$.
Let $P_n=\left\{ \left[ 0,\dfrac {1}{n}\right],\left[ \dfrac {1}{n},\dfrac {2}{n}\right] ,\ldots ,\left[ 1-\dfrac {1}{n},1\right] \right\}$ be partition of $[0,1]$. Then, $\lim _{n\rightarrow \infty } L(f,P_n)=\lim _{n\rightarrow \infty } U(f,P_n)=\dfrac {1} {2}$.
My question is how did writer find this $\lim _{n\rightarrow \infty } L(f,P_n)=\lim _{n\rightarrow \infty } U(f,P_n)=\dfrac {1} {2}$? Can you explain?
I tried something for my question:
$$\lim _{n\rightarrow \infty } L(f,P_n)=\lim _{n\rightarrow \infty } \sum ^{n}_{i=1}m_{i}\Delta _{i}= \lim _{n\rightarrow \infty } \sum ^{n}_{i=1}\sup \left\{ f\left( x\right) :x\in \left[ x_{i-1},x_{i}\right] \right\}(x_i-x_{i-1})$$
So,hence how can I find $\dfrac {1} {2}$ from this?