Question about Tagged Partitions and $\delta$-fine partitions

Question

Let $f:[a,b]\rightarrow \mathbb{R}$ where $|f|$ is bounded by $M>0$. Assume $f$ is Riemann Integrable. Let $\epsilon>0$ be arbitrary but fixed. Let $P_{\epsilon}$ be a partition of $[a,b]$ such that $U(f,P_\epsilon)-L(f,P_\epsilon)<\frac{\epsilon}{3}$. Assume $P_\epsilon$ consists of $n$ subintervals (and $n+1$ points), and set $\delta = \frac{\epsilon}{9nM}$.

Prove that if $Q$ is an arbitrary tagged partition of $[a,b]$ that is $\delta$-fine, and $P=P_\epsilon \cup Q$ is a common refinement, then $U(f,P)-L(f,P)<\frac{\epsilon}{3}$

Here is my attempt. Any scrutiny/feedback would be greatly appreciated.

Proof. Since $Q$ is $\delta$-fine, then $\max\{\Delta x_k:x\in [x_{k-1},x_{k}]\}<\delta$ for all $k=1,\dots,N$. Now since $$U(f,Q)-L(f,Q)=\sum_{k=1}^{N}(M_{k}-m_{k})\Delta x_{k}$$ $$<\frac{\epsilon}{9nM}\sum_{k=1}^{N}(M_{k}-m_{k})$$ Where $m_{k}=\inf\{f(x):x\in[x_{k-1},x_{k}]\}$ and $M_{k}=\sup\{f(x):x\in[x_{k-1},x_{k}]\}$. Since $M$ is a bound on $f$ then $M_k - m_k<2M$, so we can write $$<\frac{\epsilon}{9nM}\sum_{k=1}^{N}(M_{k}-m_{k})<\frac{\epsilon}{9nM}\sum_{k=1}^{N}2M=\frac{2N\epsilon}{9n}$$ So if $Q$ is the trivial partition consisting of only the endpoints $a$ and $b$, then $N=1$. And since $\frac{2\epsilon}{9n}<\frac{\epsilon}{3}$ (since $n\geq 2$), then $$U(f,Q)-L(f,Q)<\frac{\epsilon}{3}$$ for any choice of $P_{\epsilon}$. Therefore, since $P=P_\epsilon \cup Q$ is a common refinement of $P_\epsilon$ and $Q$, then $U(f,P)-L(f,P)<\frac{\epsilon}{3}$.

Answer 1

Your proof is flawed. You require $Q$ to be $\delta$-fine where $\|Q\| < \delta = \epsilon/(9nM)$, where $\delta$ is a variable bound that can be arbitrarily close to $0$. Then you enforce $N=1$ which implies $\|Q\| = b-a.$

The proof of your statement is trivial. Since $U(P_\epsilon,f) - L(P_\epsilon,f) < \frac{\epsilon}{3} $ and $P = P_\epsilon \cup Q$ is a refinement of $P_\epsilon$, we have $L(P_\epsilon,f) \leqslant L(P,f) \leqslant U(P,f) \leqslant U(P_\epsilon,f)$, and it follows that

$$U(P,f) - L(P,f) < \frac{\epsilon}{3}.$$

There is a useful lemma for proving the equivalence of Riemann integration defined in terms of partition refinement and in terms of $\delta$-fine partitions.

Given the existence of a partition $P_\epsilon$ such that $U(P_\epsilon,f) - L(P_\epsilon,f) < \epsilon$, there exists $\delta > 0$ such that for any $\delta$-fine partition $P$ we have $U(P,f) - L(P,f) < \epsilon$.

Perhaps this is what you intended to prove. See here for the type of argument used to prove this.