Constructing Rat./Irrat. Seqs. which Conv. to Any Real Number

# Constructing Rational/Irrational Sequences which Converge to Any Real Number

 Theorem 1: For any $x \in \mathbb{R}$ there exists a sequence $(r_n)$ of rational numbers which converges to $x$.
• Proof: Let $x \in \mathbb{R}$. For each $n \in \mathbb{N}$ let:
(1)
\begin{align} \quad I_n = \left ( x - \frac{1}{n}, x + \frac{1}{n} \right ) \end{align}
• By the density of the rational numbers, each of these nonempty intervals $I_n$ contain a rational number. For each $n \in \mathbb{N}$ let $r_n \in I_n$ be any of the rational numbers in this interval. Then we claim that $(r_n)$ converges to $x$.
• To show this, let $\epsilon > 0$ be given. Then choose $N \in \mathbb{N}$ such that $\frac{1}{N} < \epsilon$. Then if $n \geq N$ we have that $\frac{1}{n} \leq \frac{1}{N} < \epsilon$ and so:
(2)
\begin{align} \quad |r_n - x | < \frac{1}{n} \leq \frac{1}{N} < \epsilon \end{align}
• Therefore $(r_n)$ converges to $x$ and by construction, $(r_n)$ is a sequence of rational numbers. $\blacksquare$
 Theorem 2: For any $x \in \mathbb{R}$ there exists a sequence $(q_n)$ of irrational numbers which converges to $x$.
• Proof: The proof is analogous to that of theorem 1. $\blacksquare$