This is the question below :
Problem.1) Prove that if $f : [a, b] \to \mathbb{R}$ is a bounded function, then $$ \underline{\int_{a}^{b}} f(x) \, \mathrm{d}x \leq \overline{\int_{a}^{b}} f(x) \, \mathrm{d}x. $$
This is my attempt at the question, I would like to get some guidance on how to go about doing this proof and the corresponding notation.
Attempted Solution.2)
$$ \text{For any partition } \Delta x = x_{i} - x_{i-1} $$
$$ \inf\left(f(x)\right) \leq \sup\left(f(x)\right) \ : \ x \in [x_{i-1}, x_{i}] $$
$$ m_{i} = \inf\{ f(x) : x \in [x_{i-1}, x_{i}]\} \leq M_i = \sup\{ f(x) : x \in [x_{i-1}, x_{i}]\} $$
$$ L(P, f) = \sum_{i=1}^{n} m_{i} \Delta x_{i} \leq U(P, f) = \sum_{i=1}^{n} M_{i} \Delta x_{i} $$
$$ \underline{\int_{a}^{b}} f(x) \, \mathrm{d}x = \sup\{ L(P, f) : P \in \mathcal{P}[a, b]\} \leq \overline{\int_{a}^{b}} f(x) \, \mathrm{d}x = \inf\{ U(P, f) : P \in \mathcal{P}[a, b]\}, $$
where $\mathcal{P}$ is a partition.
$$ \underline{\int_{a}^{b}} f(x) \, \mathrm{d}x \leq \overline{\int_{a}^{b}} f(x) \, \mathrm{d}x. $$
From the editor. Some typos are fixed by the editor, such as incorrect use of inf/sup and upper/lower sums, as well as sudden change in fonts.
Your first steps show that $L(P,f) \leqslant U(P,f)$, where the same partition $P$ is used in the lower and upper sum. To finish you need to show that for (different) partitions $P$ and $Q$ we also have
$$\tag{*}L(P,f) \leqslant U(Q,f)$$
It then would follow that with $Q$ fixed,
$$\sup_P L(P,f) \leqslant U(Q,f),$$
and, subsequently,
$$\sup_P L(P,f) \leqslant \inf_QU(Q,f)$$
To prove (*) take a common refinement $R = P \cup Q$ and show that we must have
$$L(P,f) \leqslant L(R,f) \leqslant U(R,f) \leqslant U(Q,f)$$
I'll leave that to you with the hint that you should consider what happens to the ordering of lower and upper sums when a single new point is added to a partition.