In Wikipedia, the entry Dirichlet's approximation theorem states as follows:
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real number $\alpha$ and any positive integer $N$, there exists integers $p$ and $q$ such that $1 \leq q \leq N$ and $${\displaystyle \left|q\alpha -p\right|<{\frac {1}{N}}} .\tag1$$ This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational $\alpha$, the inequality $$ \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}}}\tag2$$ is satisfied by infinitely many integers $p$ and $q$.
Let $(1)$ be divided by $q$. We have $$ \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{qN}}\leq \frac{1}{q^2},$$ which is just $(2)$. But my question is that:
(1) how to know that there exist infinitely many $p$ and $q$ for a given irrational $\alpha$?
(2) if $\alpha$ is rational, is the number of such $p$ and $q$ finite?
(3) if there exist infinitely many $p$ and $q$ for a given irrational $\alpha$, can we choose a sequence $q_1,q_2,q_3,\cdots,q_n,\cdots $ such that $q_n \to +\infty$?