A question about Jordan measure. 
(a) Suppose that $A \subset [a,b]$ and that exists a partition $P$ of $[a,b]$ such that $c_e(A;P)<\eta$. Show that  exists $\delta>0$ such that, if $Q$ is a partition where $|Q|<\delta$ so $c_e(A;Q)<\eta$
(b)Suppose that $A \subset [a,b]$ and that exists a partition $P$ of $[a,b]$ such that $c_i(A;P)>\eta$. Show that exists $\delta>0$ such that, if $Q$ is a partition where $|Q|<\delta$, so $c_i(A;Q)>\eta$.
(c) Show that, if $A\subset [a,b]$ and $\{P_m\}$ are a sequence of partitions of $[a,b]$ such that $|P_m| \rightarrow 0$, so
  $$c_e(A)=\lim_{n\rightarrow+\infty}{c_e(A;P_m)}$$$$c_i(A)=\lim_{n\rightarrow+\infty}{c_i(A;P_m)}$$
  (d) Show that: 
  $$c_e(A)=c_i(A)+c_e(\partial A)$$

DEFINITIONS AND ADDITIONAL INFORMATION
Def.0: $P$ a partition, so $|P|=\max\{x_{i+1}-x_i: x_i, x_{i+1}\in P\}$
Def.1: $A \subset [a,b]$ is a negligible if, for all $\varepsilon>0$ exists partition $P=\{x_0,...,x_n\}$ of $[a,b]$ such that $\sum_P^*\Delta x_i<\varepsilon$, where this sum are over the indices such that $A \cap [x_{i-1},x_i] \neq \emptyset$
Follows from an exercise in class that: 
$A\subset[a,b]$ is negligible is the same that affirm that: for all $\varepsilon>0$ exists $\delta>0$ such that, for all partition $Q$ where $|Q|<\delta$, we have
$$\sum_{Q}^* \Delta x_i<\varepsilon$$, where $\sum^*_Q$ denote the sum over the indices such that $A \cap [x_{i-1},x_i] \neq \emptyset$
In this exercise we will generalize this result.
Def.2: $c_e(A;P)\dot{=}\sum_{I_k \cap A \neq \emptyset}  c(I_k)=\sum_P^* \Delta x_i$
Def.3: $c_e(A)=\inf\{c_e(A;P):P\in\textrm{Part}([a,b])\}$
Def.4: $c_i(A;P)\dot{=}\sum_{I_k \subset A} c(I_k)$
Def.5: $c_i(A)=\lim_{n\rightarrow+\infty}{c_i(A;P_m)}$
Obs.: I'm using that: $P=\{x_0,...,x_n\}$ and $Q=\{y_0,...,y_m\}$.
COMMENTS: This is a long exercise of my course or Measure and Integration. I did but I'm afraid of being too informal in some way. I need someone to look at my answer (especially the item d) and help me to write more clearly my reasoning and correct any possible mistake, I feel that this bad.
MY ATTEMPT
(a)
Let $0<\delta<|P|$, $|Q|<\delta$, $Q=\{y_0,...,y_m\}$ and define:
            $$
   \Lambda_1= \{ i \in \{1,...,m\}: [y_{i-1}, y_i] \subset [x_{j-1}, x_j] \textrm{ for some } j\in \{1,...,n\} \}
   $$
            $$
   \Lambda_2= \{ i \in \{1,...,m\}: [y_{i-1}, y_i] \cap \{x_{j-1}, x_j\} \textrm{ for some } j\in\{1,...,n\}  \neq \emptyset\}
   $$
            So, 
            $$
   c_e(A;Q)=\sum_{I_{k'} \cap A \neq \emptyset}  c(I_{k'})=\sum_Q^* \Delta y_i=\sum_{\Lambda_1}^* \Delta y_i+\sum_{\Lambda_2}^* \Delta y_i
   $$
            Define, 
            $$
   \Lambda^1_1= \{ i \in \{1,...,n\}: [y_{j-1}, y_j] \subset [x_{i-1}, x_i] \textrm{ for some } j\in \{1,...,m\} \}
   $$
            $$
   \Lambda^1_2= \{ i \in \{1,...,n\}: [y_{j-1}, y_j] \cap \{x_{i-1}, x_i\} \textrm{ for some } j\in\{1,...,m\}  \neq \emptyset\}
   $$
            So, 
            $$
   c_e(A;P)=\sum_{I_{k} \cap A \neq \emptyset}  c(I_{k})=\sum_P^* \Delta y_i=\sum_{\Lambda_1^1}^* \Delta y_i+\sum_{\Lambda_2^1}^* \Delta y_i
   $$
            Note that:
            $$
   \sum_{\Lambda_1}^* \Delta y_i <\sum_{\Lambda_1^1}^* \Delta y_i
   $$
            because each $[y_{j-1},y_j] \subset [y_{i-1},y_i]$ and 
            $$
   \sum_{\Lambda_2}^* \Delta y_i <\sum_{\Lambda_1^2}^* \Delta y_i
   $$
            because $|Q|<\delta$. So, $c_e(A,Q)<c_e(A,P)=\eta$
(b)Let $0<\delta<|P|$, $|Q|<\delta$, $Q=\{y_0,...,y_m\}$. As $\{ \bigcup [x_{i-1}, x_i]:  [x_{i-1}, x_i] \subset A\} \subset \{ \bigcup [y_{j-1}, y_j]:  [y_{j-1}, y_j] \subset A\}$ follows that
            $$
   \eta<c_i(A;P)<c_i(A;Q)
   $$
(c) Note that, for all partition $P$ such that $\eta>c_e(A;P)>0$,let $|P|>\delta>0$ and $0<|Q|<\delta \Rightarrow c_e(A,Q)<\eta$, and using "(a)",  $c_e(A;Q)<\eta$. So we can write that $P_m$ with $|P_m| \rightarrow 0$, exists subsequence $P_{i_k}$ such that for all $i_k$
            $$
   c_e(P_{ik})>c_e(P_{i_k+1})
   $$
            So, $ \lim_{n\rightarrow+\infty}{c_e(A,P_m)}=\inf\{c_e(A;P):P\in\textrm{Part}([a,b])\}=c_e(A)$
Now note that, using "(b)", for all partition $P$ such that $c_i(A;P)>\eta>0$, let $|P|>\delta>0$ and $0<|Q|<\delta \Rightarrow c_i(A,Q)>\eta$. So we can write that $P_m$ where $|P_m| \rightarrow 0$, exists subsequence $P_{i_k}$ such that for all $i_k$
            $$
   c_i(P_{ik})<c_i(P_{i_k+1})
   $$
            So, $ \lim_{n\rightarrow+\infty}{c_i(A,P_m)}=\sup\{c_e(A;P):P\in\textrm{Part}([a,b])\}=c_i(A)$
(d)$$ c_e(A)-c_e(\partial A)=\lim_{n\rightarrow +\infty}{c_e(A;P_m)}-\lim_{n \rightarrow +\infty}c_e(\partial A;P_m)=$$
            $$
   =\lim_{n\rightarrow+\infty}{c_e(A \backslash \partial A;P_m)}=\lim_{n\rightarrow+\infty} c_i(A,P_m)=c_i(A)
   $$
 A: Suppose the partition $P $ has $n$ intervals.
a. it suffices to show that for each $\epsilon > 0 $ there exists $\delta > 0$ such that
$$|Q| < \delta \Longrightarrow c_e(A;Q) \leq c_e(A;P) + \epsilon.$$
Let $\epsilon$ be given, define $\delta$ such that $(n-1)\delta < \epsilon.$
Now suppose $I_q\in Q$ with $I_q \cap A \neq \emptyset$, then either 


*

*$I_q\subset I_p$ for some $I_p \in P$ with $I_p \cap A\neq \emptyset$, or

*$I_q^*$ contains at least one of the $n-1$ cut points in the partition $P$.
$\sum |I_q| \leq \sum |I_p|$ and $\sum |I_q^*| < \epsilon$.

b.  it suffices to show that for each $\epsilon > 0 $ there exists $\delta > 0$ such that
$$|Q| < \delta \Longrightarrow c_i(A;Q) \geq c_i(A;P) - \epsilon.$$
Let $\epsilon$ be given, again define $\delta$ such that
$(n-1)\delta < \epsilon$.

c. 
$$c_e(A) \leq c_e(A,P_n) \text{ for all } n\in \mathbb{N}.$$ 
By the definition of infinium and (a), for each $\epsilon > 0$ there exists $P^*$ and $N\in \mathbb{N}$ such that $\forall n > N$ we have 
$$c_e(A) + \epsilon \geq c_e(A, P^*) +\frac{\epsilon}{2} \geq c_e(A,P_n) .$$
The proof for inner Jordan measure is very similar.
