In From Real to Complex Analysis, there is an exercise that is stated as follows:
Using merely the definition of integrability, show that the function $f:[0,1]\to\mathbb{R}$ defined by $f(x)=x^2$ for all $x\in [0,1]$ is Riemann-integrable on $[0,1]$ and that $\int_0^1f=1/3$.
I am able to show that there exists a partition $P$ of $[0,1]$ such that $U(P,f)=L(P,f)=1/3$ (Namely, let $P$ be the limit of $(P_n)$ as $n\to\infty$, where $P_n=\{0,1/n,2,n,...,1\}$). If I were able to use Darboux's theorem, this would be enough. However, it says to only use the definition of integrability. So, I have to show that $U(P,f)\geq 1/3\geq L(P,f)$ for all partitions $P$ of $[a,b]$. There is a lemma that I could use, which states that, when $P,Q$ are partitions of $[a,b]$ and $Q$ is a refinement of $P$ containing $k$ points in addition to those of $P$, and $|f(x)|<K$ for all $x\in [a,b]$, $0\leq U(P,f)-U(Q,f)\leq 2kKw(P)$ and $0\leq L(Q,f)-L(P,f)\leq 2kKw(P)$ (where $w(P)$ is the width of $P$). However, again, it says to only use the definition of integrability. The only way I can really think to do this is to show that there are elements of the aforementioned sequence $(P_n)$ that come arbitrarily close to a refinement (call it $R$) of $P$ for any $P$, so $R$ must be such that $U(R,f)\geq 1/3\geq L(R,f)$, so $U(P,f)\geq 1/3\geq L(P,f)$. However, this uses the fact that the upper sum is decreasing on refinement, and the lower sum is increasing on refinement, which is a corollary of the above lemma. I could prove this as part of my solution, but it feels wrong to simply prove the results I want to use, that the question has told me not to use, in my solution. That feels like cheating.
I tried plugging the function into the formula for the upper and lower sums, but all I get is a very difficult inequality I have to prove. For the upper sum, it's a special case of showing that, if $0<x_1<x_2<\cdots<x_n$, then $\sum_{j=1}^{n-1}x_j(x_{j+1}+x_j)(x_{j+1}-x_j)<\frac{2}{3}x_n^3$, which clearly isn't what the authors want me to be thinking about. I'm not really asking for help with the maths, unless there is a simple way of doing this that I'm missing; rather, I want to know if, when they set this problem, the authors likely expected me to use the lemma I mentioned, or prove that the upper sum is decreasing, and the lower sum increasing, on refinement. Both of these count as using established results other than the definition of integrability to me, but I don't see how they could expect me to solve it any other way.
Notice for any $0 \le x < y$, we have $$(y-x)y^2 > \frac13(y-x)(y^2 + xy + x^2) = \frac13(y^3-x^3)$$ For any partition $P: 0 = x_0 < x_1 < \ldots < x_n = 1$ of $[0,1]$, substitute $(x,y)$ by $(x_{i-1},x_i)$ and sum over $i$, one obtain: $$U(f,P) = \sum_{i=1}^n (x_{i}-x_{i-1})x_i^2 > \frac13 \sum_{i=1}^n (x_i^3 - x_{i-1}^3) = \frac13 (x_n^3 - x_0^3) = \frac13$$