I need to prove $U_{|f |} (Z ) − L_{|f |} (Z ) ≤ U_f (Z ) − L_f (Z )$
Where $U$ is the upper estimate and $L$ is the lower estimate and $f$ is bounded.
I get that since $f$ is bounded, I have $m_j = inf(f)$ and $M_j = sup(f)$
Since $|f| ≥ f$, there is also $q_j = inf(|f|)$ and $Q_j = sup(|f|)$ and $q_j ≥ m_j, Q_j ≥ M_j$
So then $U_{|f |} ≥ U_f (Z )$ and $L_{|f |} (Z ) ≥ L_f (Z )$
Which gives me two equalities and something like this when combined: $$U_{|f |} (Z ) + L_{|f |} (Z ) ≥ U_f (Z ) + L_f (Z )$$ which is quite off.
How can I prove $U_{|f |} (Z ) − L_{|f |} (Z ) ≤ U_f (Z ) − L_f (Z )$?
!read this answer after the comment i just put.
if $M_j \geq 0$ then $Q_j=M_j$ , and since $q_j \geq m_j$: $$ Q_j-q_j \leq M_j - m_j \tag{1} $$ if $M_j <0$ then $f(x)$ is strictly negative on $\Delta x_j$, hence $|f(x)| = -f(x)$ therefore $Q_j = -m_j$ and $q_j = - M_j$ so : $$ Q_j -q_j = -(m_j-M_j)=M_j-m_j. \tag{2} $$ for every $j$ either $M_j \geq0$ or $M_j<0$ and in both case $Q_j-q_j \leq M_j - m_j$. so multiply this by length of $\Delta x_j$ and sum over all $j$: $$ \sum_j (Q_j-q_j)\Delta x_j \leq \sum_j (M_j-m_j)\Delta x_j $$ it's true for every partition $P$ and then take $||P|| \rightarrow 0$.