Rational approximation of irrational number

[Grigorios Kostakos]

Show that for every irrational number \(\alpha\) and for every positive integer \(n\) there exist an integer \(p_n\) and a positive integer \(q_n\) such that \[\biggl|{\alpha-\frac{p_n}{q_n}}\biggr|<\frac{1}{n\,q_n}\,.\] Show that the integers \(p_n\) and \(q_n\) can be chosen in such a way that we have \[\biggl|{\alpha-\frac{p_n}{q_n}}\biggr|<\frac{1}{q_n^2}\,.\]