If the Partition $P\subset P'$, then $L(f, P)\leq L(f, P')$ and $U(f, P))\geq U(f, P')$

Question

I have this proof but I don't really understand it, mathematically and visually (if that makes sense).

So the definition is:

A partition $P'$ is a refinement of the partition $P$ is $P\subset P'$.

And the Lemma is:

If $P\subset P'$, then $L(f, P)\leq L(f, P')$ and $U(f, P)\geq U(f, P')$

Here's the Proof:

a) Let $P$ equal {$x_0, x_1,...,x_n$} and $P'$ equal {$x_0, x_1, x_2,...,x_{j-1}, \gamma, x_j,...,x_n$}

Then $L(f, P)=\sum_{i=1}^{n}\inf_{[x_{i-1}, x_i]} f(x)\cdot(x_i-x_{i-1})$ $=\sum_{i=1\neq j}^{n}\inf_{[x_{i-1}, x_i]} f(x)(x_i-x_{i-1})+\inf_{[x_{j-1}, x_j]}f(x)(x_j-\gamma +\gamma-x_{j-1}) \rightarrow (1)$

Now, $\inf_{[x_{j-1}, x_j]}f(x)=\inf${$f(x) \mid x\in[x_{j-1}, x_j]$} $\leq \inf${$f(x)\mid x \in{[x_{j-1}}, \gamma]$} and

$\leq\inf${$f(x)\mid x\in[\gamma, x_j]$} $=\inf_{[\gamma, x_j]}f(x)$

$(1)\leq\sum_{i=1\neq j}^{n}\inf_{[x_{i-1},x_i]}f(x)(x_i-x_{i-1})+\inf_{[x_{j-1}, \gamma]}f(x)(\gamma-x_{j-1})+\inf_{[\gamma, x_j]}f(x)(x_j-\gamma)=L(f, P')$

Analogously, $U(f,P)\geq U(f, P')$

b) If $P'$ differs from $P$ by $m$ points, repeat the procedure in a) $m$ times.

QED

NOTE

So I understand the concept of partitions, but this proof confuses the hell out of me. 1. What is $P'$ and why is it necessary to consider. 2. What is the lemma actually saying (visually) 3. I don't get the procedure after the second line. ANY help would GREATLY be appreciated, thanks!! :)

Answer 1

1. Riemann sums $\sum_{k=1}^N f(\xi_k)(x_k-x_{k-1})$ (or similar) are finite sums referring to a chosen partition ${\cal P}$ of the interval $[a,b]$. But the integral $\int_a^b f(x)\>dx$ is a limit of such sums when ${\cal P}$ is refined to oblivion (it's more complicated than $n\to\infty$). In order to see what happens "in the limit" we have to study what happens to such a Riemann sum when a partition ${\cal P}$ is refined to a partition ${\cal P}'$ by introducing additional separation points.

2. Upper sums clearly overshoot the intended limit, but there is hope: Under refinement of ${\cal P}$ the upper sum decreases, hence the sum computed with the finer partition will be a better approximation to the intended limit. Similarly: Under a refinement the lower sum will increase.

3. Forget about the machinery set up for the proof of the lemma. The basic idea is very simple indeed.

Assume that an interval $J$ of the partition is split into two intervals $J'$ and $J''$ by an additional separation point. Then $|J|=|J'|+|J''|$, where I have written $|\cdot|$ for the length of these intervals. It follows that $$\sup_J f\cdot |J|=\sup_J f\cdot \bigl(|J'|+|J''|)\geq\sup_{J'}f\cdot |J'|+\sup_{J''} f\cdot |J''|\ ,\tag{1}$$ because $\sup f$ over the large interval $J$ tends to be larger than $\sup f$ over the subintrvals $J'$, $J''$. In $(1)$ the LHS is the contribution of $J$ in $U(f,{\cal P})$, and the RHS is the total contribution of $J$ in $U(f,{\cal P}')$.

Draw a figure!