Sorry for the wonky title -- I had to really condense the exercise to fit all the LaTeX code in! Anyways, I am working on Exercise 7.1.3 from Lesson 7.1 in Real Analysis by Cesar O. Aguilar. The exercise says:
If $f$ is Riemann integrable on $[a,b]$ and $(\dot{\mathcal{P}}_n)$ is a sequence of tagged partitions of $[a,b]$ such that $||\dot{\mathcal{P}}_n||\to 0$ prove that $$ \int_a^b f=\lim_{n\to\infty}S(f;\dot{\mathcal{P}_n}).$$
I'm wondering if my attempt below is correct. I'm particularly unsure about my choice of $\delta_1$ and $\delta$.
Proof: Since $||\dot{\mathcal{P}}_n||\to 0,$ given $\delta_1>0,$ there exists $N\in\mathbb{N}$ such that if $n\geq N,$ we have that $||\dot{\mathcal{P}}_n||<\delta_1.$ Now suppose $\epsilon>0$. Then since $f\in\mathcal{R}[a,b],$ given any tagged partition $\dot{\mathcal{P}}$ of $[a,b],$ there exists some $\delta_2>0$ such that if $||\dot{\mathcal{P}}||<\delta_2$, then $\left|S(f;\mathcal{P})-\int_a^b f\right|<\epsilon$. Let $\delta=\min(\delta_1,\delta_2).$ Then if $n\geq N$ and $||\dot{\mathcal{P}}_n||<\delta,$ we have that $\left|S(f;\dot{\mathcal{P}}_n)-\int_a^b f\right|<\epsilon.$ Therefore, $$ \int_a^b f=\lim_{n\to\infty}S(f;\dot{\mathcal{P}_n}).$$
$\blacksquare$