# Help for this problem involving rieman integral and partitions

If $$f: I--->\mathbb R$$ is bounded, let $$||f||:= Sup {|f(x)|: x \in I}$$, and if $$P =(x_0,...,x_n)$$ is a partition of $$I:=[a,b]$$, let $$||P||:=Sup [x_1-x_0,...,x_n-x_{n-1}]$$

(a) If $$P'$$ is the partition obtained from $$P$$ as in the proof of Lemma 7.1.2, show that $$L(P,f) ≤ L(P',f) ≤ L(P,f) + 2||f|| ||P||$$ and $$U(P,f) ≥ U(P',f) ≥ U(P,f) - 2||f|| ||P||$$

(b) If $$P_1$$ is a partitions obtained from $$P$$ by adding $$k$$ points to $$P$$, show that $$L(P_1,f) ≤ L(P,f) + 2k||f|| ||P||$$ and also that $$U(P_1,f) ≥ U(P,f) - 2k||f|| ||P||$$

So what i have intuitively and some hint of my professor is that

First, the partition of Lemma 7.1.2 is

$$P':=[{x_0,x_1,...,x_{k-1},z,x_k,...,x_n}$$]

So what i have so far is, and part of this, is a hint of the professor we have

$$|m_k|$$, $$|m'_k|$$, $$|m''_k|$$ $$≤$$ $$||f||$$ hence $$0 ≤ L(P',f) - L(P,f) = (m'_k-m_k)(z-x_{k-1})+(m''_k-m_k)(x_k-z)$$ $$≤$$ $$2||f||(x_k-x_{k-1} ≤ 2||f|| ||P||$$

Where

$$m_k:=inf[f(x):x \in [x_{k-1},x_k]]$$

$$m'_k:=inf[f(x):x \in [x_{k-1},z]]$$

$$m''_k:=inf[f(x):x \in [x_z,x_k]]$$

And for part b i think i need to use induction but i think i need to solve a first to solve b and i... im lost sincerely ): can someone help me?

• You seem to have concluded $L(P,f) ≤ L(P',f) ≤ L(P,f) + 2||f|| ||P||$, and $U(P,f) ≥ U(P',f) ≥ U(P,f) - 2||f|| ||P||$ follows similar logic. So what are you confused with for (a)? – Acccumulation Nov 20 '18 at 23:07
• Because this is only a partial solution to (a) and i dont know the dateails or why is that inequality true – Daniel ML Nov 21 '18 at 1:08