So the statement we are tasked with proving is as follows
If $f,g\in\mathbb{B}[a,b]$ are such that $f(x)\neq g(x)$ for only a finite number of points, $x\in[a,b]$, then $$ \underline{\int_a^b}f~dx=\underline{\int_a^b}g~dx~~~\mathrm{and}~~~\overline{\int_a^b}f~dx=\overline{\int_a^b}g~dx $$ And, in particular, $f\in\mathbb{R}[a,b]\iff g\in\mathbb{R}[a,b]$
Now, I've seen this post, and used the idea provided in the answer to prove the theorem for myself. I found, however, that this is one such proof where, if you're not meticulous, it'll come back to haunt you. Here is my attempt.
We start by letting $\langle P_n\rangle$ be a limiting sequence of partitions, $P_n\in\mathcal{P}[a,b]$, $\forall n\in\mathbb{N}$, of the interval $[a,b]$. Suppose also that $\gamma_1,\gamma_2,\cdots,\gamma_k\in[a,b]$ are the values in $[a,b]$ at which $f(x)\neq g(x)$. Suppose, in addition, that $\gamma_1<\gamma_2<\cdots<\gamma_k$. Define now $$ Q=\min_{1\leq i\leq k}(\gamma_i-\gamma_{i-1}) $$ Now we note that, since $\langle P_n\rangle$ is a limiting sequence, we know that $\exists N\in\mathbb{N}$ such that $n>N\implies ||P_n||<Q$. Suppose therefore that $n>N$, and hence consider now the expression $\mathcal{L}(f,P_n)-\mathcal{L}(g,P_n)$. We note that any interval $[x_{i-1},x_i]$, $x_{i-1},x_i\in P_n$, contains at most one value $\gamma\in\lbrace\gamma_1,\gamma_2,\cdots,\gamma_k\rbrace$ (since $||P_n||<Q$). Therefore, if we suppose that each $\gamma_j$ is contained in $[x_{j-1},x_j]$, for some $x_{j-1},x_j\in P_n$, then we have that, if $$ m_i=\inf_{x\in[x_{i-1},x_i]}f(x)~~~\mathrm{and}~~~w_i=\inf_{x\in[x_{i-1},x_i]}g(x) $$ then $m_j-w_j=\min\lbrace0,f(\gamma_j)-g(\gamma_j)\rbrace$, $1\leq j\leq k$. Consequently, we have that \begin{equation*} \begin{split} \mathcal{L}(f,P_n)-\mathcal{L}(g,P_n) & =\sum_{j=1}^k\min\lbrace0,f(\gamma_j)-g(\gamma_j)\rbrace(x_j-x_{j-1}) \\ & \leq\sum_{j=1}^k\min\lbrace0,f(\gamma_j)-g(\gamma_j)\rbrace||P_n|| \\ & =||P_n||\sum_{j=1}^k\min\lbrace0,f(\gamma_j)-g(\gamma_j)\rbrace \\ & \to0\sum_{j=1}^k\min\lbrace0,f(\gamma_j)-g(\gamma_j)\rbrace \\ & =0 \end{split} \end{equation*} That is, $\mathcal{L}(f,P_n)-\mathcal{L}(g,P_n)\to0$, as $n\to\infty$. Consequently, we have that \begin{equation*} \begin{split} \lim_{n\to\infty}(\mathcal{L}(f,P_n)-\mathcal{L}(g,P_n)) & =0 \\ \lim_{n\to\infty}\mathcal{L}(f,P_n)-\lim_{n\to\infty}\mathcal{L}(g,P_n) & =0 \\ \underline{\int_a^b}f(x)~dx-\underline{\int_a^b}g(x)~dx & =0 \\ \underline{\int_a^b}f(x)~dx & =\underline{\int_a^b}g(x)~dx \\ \end{split} \end{equation*} As required for the first part. For our purposes, this completes the proof. It should be noted that the second half of the proof is very similar.
My main concern with this proof is style. I think it reads rather poorly, and so if anyone can provide me with some stylistic tips, I would greatly appreciate it. Of course, if the proof lacks validity, please do also point this out to me. Thank you in advance for any feedback.