Let $\alpha \in \mathbb { R } _ { + }$. We assume that there exist two sequences of non-zero natural integers $\left( p _ { n } \right) _ { n \in \mathbb { N } }$ and $\left( q _ { n } \right) _ { n \in \mathbb { N } }$ such that $$\lim _ { n \rightarrow + \infty } \frac { p _ { n } } { q _ { n } } = \alpha \quad \text { and } \quad \left| \alpha - \frac { p _ { n } } { q _ { n } } \right| \underset { n \rightarrow + \infty } { = } o \left( \frac { 1 } { q _ { n } } \right)$$ We further assume that for all $n \in \mathbb { N } , \frac { p _ { n } } { q _ { n } } \neq \alpha$. Show that $\alpha$ is an irrational number.
Let $\alpha \in \mathbb { R } _ { + }$. We assume that there exist two sequences of non-zero natural integers $\left( p _ { n } \right) _ { n \in \mathbb { N } }$ and $\left( q _ { n } \right) _ { n \in \mathbb { N } }$ such that
$$\lim _ { n \rightarrow + \infty } \frac { p _ { n } } { q _ { n } } = \alpha \quad \text { and } \quad \left| \alpha - \frac { p _ { n } } { q _ { n } } \right| \underset { n \rightarrow + \infty } { = } o \left( \frac { 1 } { q _ { n } } \right)$$
We further assume that for all $n \in \mathbb { N } , \frac { p _ { n } } { q _ { n } } \neq \alpha$.
Show that $\alpha$ is an irrational number.