1. I need some suggestion

on how to prove this problem. I don't know where to start. Any help is very much appreciated.

Suppose that a function f: [a,b] >> R is bounded and P = {x_0,x_1,...,x_n} is a partition of [a,b]. Then there exist 2 real numbers m and M such that

m(b-a) <= L(P,f) <= S(P,f) <= U(P,f) <= M(b-a)

for any c_k in [x_k-1,x_k]

2. Originally Posted by EmmWalfer
on how to prove this problem. I don't know where to start. Any help is very much appreciated.

Suppose that a function f: [a,b] >> R is bounded and P = {x_0,x_1,...,x_n} is a partition of [a,b]. Then there exist 2 real numbers m and M such that

m(b-a) <= L(P,f) <= S(P,f) <= U(P,f) <= M(b-a)

for any c_k in [x_k-1,x_k]
What if you took $\displaystyle M=\sup_{x\in[a,b]}f(x)$ and $\displaystyle m=\inf_{x\in[a,b]}f(x)$ which you know exist by the assumption of boundedness. Then,

\displaystyle \begin{aligned}U\left(P,f) &=\sum_{i=0}^{n}\sup_{x\in[x_{i-1},x_i]}f(x)\Delta x_i\\ &\leqslant \sum_{i=0}^{n}M \Delta x_i\\ &=M\sum_{i=0}^{n}\Delta x_i\\ &= M(b-a)\end{aligned}

See if you can figure out how to prove the others.