Denote
$$E = \left\{x\in [0, 1]: \text{there exist infinitely many }p,q \in\mathbb N \text{ such that }\left|x −\frac{p}{q}\right|\le \frac{1}{q^{3}} \right\}.$$
Show that $m(E) = 0$.
I am totally unaware how to begin this problem. Any hints are appreciated.
For each $q$, consider $$E_q = \bigcup_{p=0}^{q} \left( \frac pq - \frac{1}{q^3}, \frac pq + \frac{1}{q^3}\right).$$
Note that $m(E_q) \le \frac{q+1}{q^3} \le 2 \frac{1}{q^2}$.
From the definition of $E$, $E \subset \cup_{q= K}^\infty E_q$ for all $K$. So $$m(E) \le \sum_{q=K}^\infty m(E_q) \le 2\sum_{q=K}^\infty \frac{1}{q^2}\to 0$$ as $K \to\infty$, as $\sum \frac{1}{q^2}$ is a convergent series. Hence $m(E) = 0$.