Convergence of sequence of rationals and irrationals

Note that both \(\mathbb{Q}\) and \(\mathbb{R} \backslash \mathbb{Q}\) are dense in \(\mathbb{R}\).

1. Convergence of sequence of distinct rationals to a
• Rational number:
• Want sequence of distinct rationals say \(\{a_n\}_{n=1}^{\infty}\) to another rational number say \(a\). Choose \(a_k = a + \frac1k\) where \(k \in \mathbb{Z}^+\). Clearly, \(a_k\)'s are rationals and converge to the rational number \(a\).
• Irrational number:
• Want sequence of distinct rationals say \(\{a_n\}_{n=1}^{\infty}\) to an irrational number say \(a\). Choose \(a_k = \frac{\lfloor 10^k a \rfloor}{10^k}\) where \(k \in \mathbb{Z}^+\). Clearly, \(a_k\)'s are rationals and converge to the irrational number \(a\).
2. Convergence of sequence of distinct irrationals to a
• Rational number:
• Want sequence of distinct irrationals say \(\{a_n\}_{n=1}^{\infty}\) to a rational number say \(a\). Choose \(a_k = a + \frac{x}k\) where \(k \in \mathbb{Z}^+\) and \(x\) is any irrational number. Clearly, \(a_k\)'s are irrationals and converge to the rational number \(a\).
• Irrational number:
• Want sequence of distinct irrationals say \(\{a_n\}_{n=1}^{\infty}\) to another irrational number say \(a\). Choose \(a_k = a + \frac1k\) where \(k \in \mathbb{Z}^+\). Clearly, \(a_k\)'s are irrationals and converge to the irrational number \(a\).