Jtdylktuy

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?