Proving a lemma of integrability

Question

Suppose $f$ is bounded on $[a,b]$. Prove that if there exist sequences $P_n$ and $Q_n$ of partitions of $[a,b]$ such that $\lim U(f,P_n) = \lim L(f,Q_n) = L$, then $f$ is integrable on $[a,b]$ and $\int_a^b f(x) \,dx = L$.

The first thing that came to my mind was the Cauchy criterion for integrability. Since we have $\lim U(f,P_n) = \lim L(f,Q_n) = L$, we can assume that $U(f,P_n) - L < \varepsilon/2$ and $L(f,Q_n) - L < \varepsilon/2$. By using the trangle inequalities, we can get $U(f,P_n) - L(f,Q_n) < \varepsilon$. However, this is not $U(f,P_n) - L(f,P_n) < \varepsilon$. I do not know how to move forward.

Another method I can think of is to prove that $\lim U(f,P_n) = \lim L(f,Q_n) = L = U(f) = L(f)$ , which is the same to prove that $\lim U(f,P_n) = \lim L(f,Q_n)= \sup\{L(f,Q_n)\} = \inf\{U(f,P_n)\}$. I do not know how to move forward with this method, either.

My instinct told me that the best way to solve this problem is to use the Cauchy Criterion for integrability, but I felt really lost since we have two partitions here.

Thanks in advance.

Answer 1

For every $n\in\Bbb{N}$, \begin{align} L(f,P_n)\leq \sup\{L(f,P)\}\leq \inf\{U(f,Q)\}\leq U(f,Q_n) \end{align} Let $n\to \infty$ and use the fact that inequalities are preserved when taking limits to deduce \begin{align} L\leq \sup\{L(f,P)\} \leq \inf\{U(f,Q)\}\leq L. \end{align} This completes the proof (why? think of the definition of the integrability and the integral).

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Using the other approach, given $\epsilon>0$, as you have mentioned there is an $n\in\Bbb{N}$ such that \begin{align} U(f,Q_n)-L(f,P_n)<\epsilon. \end{align} So, all we have to do is refine the partition by setting $P=P_n\cup Q_n$ (because a refined partition makes upper sums smaller and makes lower sums bigger (I mean with weak inequalities not necessarily strict ones)). Then, \begin{align} U(f,P)-L(f,P)\leq U(f,Q_n)-L(f,P_n)<\epsilon. \end{align} This proves integrability (recall that all we have to do is find some partition which works for the $\epsilon$ bound).

Answer 2

Another way to go...

Let $R_n$ be the common refinement of $P_n$ and $Q_n$. That is if $a=p_1 < p_2 < \cdots < p_{n-1} < p_n = b$ and $a=q_1 < q_2 < \cdots < q_{n-1} < q_n = b$ are the partitions $P_n$ and $Q_n$, respectively, set $R_n = \{p_i : 1 \leq i \leq n\} \cup \{q_i : 1 \leq i \leq n\}$. (Notice that this avoids duplicates, so all intervals in $R_n$ have positive width.) Then $$ L(f,Q_n) \leq L(f,R_n) \leq U(f,R_n) \leq U(f,P_n) \text{,} $$ and you can finish by the squeeze theorem (since you know where the sequences on the left and right ends are going).