I found the next problem and I really stucked in the proof:
Let $a < b$. Suppose $P_i$ where $i = 1, 2, 3...$ is a sequence of partitions of $[a,b]$ whose size/mesh does not tend to 0. Show that there is a Riemann integrable function $f$ on [a,b] and a sequence of Riemann sums such that $S(f, P_i) \nrightarrow \smallint_a^bf$
My advances:
The main problem I found is that I'm unable to define a sequence of partitions between $[a, b]$ whose size doesn't tend to 0. I'm just unable to find a suitable sequence. ¿Any help here?
EDIT: my sub-question here is maybe more like, is there an expression for a sequence of partitions whose size doesn't tend to 0? Away from the trivial ones like the constant partition, or keeping some interval fixed?
From that point I found that the idea is intuitively easy: if some of the intervals of the partition does not tend to 0, then that part of the Riemman sum will be dominated by the value $f(\xi_i)$ (for some $\xi$ chosen in the interval) which is not reflecting the real variation of the function on that interval, and obviously the sum will not converge to the real integral.
Without loss of generality, consider the interval $[a,b] = [0,1]$.
Let $(P_n)$ be a sequence of partitions of $[0,1]$ such that the mesh $\|P_n\| \not \to 0$. The partition points of $P_n$ are denoted as $0 = x_0^{(n)} < x_1^{(n)} < \ldots < x_{m_n}^{(n)} = 1$.
Take the Riemann integrable function $f: x \mapsto x$. If the sequence of Riemann sums converges as $S(P_n,f) \to \int_0^1f(x) \, dx = 1/2$, then for any $\epsilon > 0$, there exists $N(\epsilon) \in \mathbb{N}$ such that for all $n \geqslant N(\epsilon)$ we have $|S(P_n,f) - 1/2| < \epsilon$. In particular, for the sequence of Riemann sums using the right endponts of subintervals as tags we would have for all $n \geqslant N(\epsilon)$,
$$\tag{1}S(P_n,f) = \sum_{j=1}^{m_n}x_j^{(n)}(x_j^{(n)} -x_{j-1}^{(n)}) < \frac{1}{2} + \epsilon$$
However, since $\|P_n\| \not\to 0$ it follows that there exists $\epsilon_0 > 0$ such that for any $N \in \mathbb{N}$ there exists $k_N \geqslant N$ such that
$$\|P_{k_N}\| = \max_{1 \leqslant j \leqslant m_{k_N}}(x_j^{(k_N)} - x_{j-1}^{(k_N)}) > \epsilon_0,$$
and this implies that there exists $j_{k_N}$ such that $x_{j_{k_N}}^{(k_N)} - x_{j_{k_N}-1}^{(k_N)} > \epsilon_0$.
We have
$$\tag{2}S(P_{k_N},f) = x_{j_{k_N}}^{(k_N)}(x_{j_{k_N}}^{(k_N)} - x_{j_{k_N}-1}^{(k_N)}) + \sum_{j \neq j_{k_N}}x_j^{(k_N)}(x_j^{(k_N)} -x_{j-1}^{(k_N)})$$
Since $f(x) = x$ is increasing, it follows that the sum on the RHS of (2) overestimates the sum of the integrals over the intervals $[0, x_{j_{k_N}-1}^{(k_N)}]$ and $[x_{j_{k_N}}^{(k_N)},1]$. Thus, for any $N \in \mathbb{N}$, there exists $k_N \geqslant N$ such that
$$S(P_{k_N},f) > x_{j_{k_N}}^{(k_N)}(x_{j_{k_N}}^{(k_N)} - x_{j_{k_N}-1}^{(k_N)})+ \int_0^{x_{j_{k_N}-1}^{(k_N)}}x \, dx+ \int_{x_{j_{k_N}}^{(k_N)}}^1x \, dx\\ =x_{j_{k_N}}^{(k_N)}(x_{j_{k_N}}^{(k_N)} - x_{j_{k_N}-1}^{(k_N)})+\frac{1}{2}[x_{j_{k_N}-1}^{(k_N)}]^2 + \frac{1}{2} - \frac{1}{2}[x_{j_{k_N} }^{(k_N)}]^2 \\ = \frac{1}{2}+ \frac{1}{2}(x_{j_{k_N}}^{(k_N)}- x_{j_{k_N}-1}^{(k_N)})^2 > \frac{1}{2} + \frac{\epsilon_0^2}{2}$$
This contradicts (1) when $\epsilon = \epsilon_0^2/2$ is chosen. Therefore, there is a Riemann integrable function $f$ and a sequence of Riemann sums such that
$$S(P_n,f) \not\to \int_0^1 f(x) \, dx$$