# Irrationality from a Limit Lemma

### Lemma

Let $\delta\in\mathbb{R}^{+}$. Assume there is a sequence of distinct positive rational numbers $r_{n}/s_{n}$ (written in reduced form) such that

$\displaystyle s_{n}\left|\delta-\frac{r_{n}}{s_{n}}\right|\rightarrow 0$,

as $n\rightarrow\infty$. Then $\delta$ is irrational.

### Proof

Assume $\delta=p/q$ in reduced form. There is at most one index $n=n_{0}$, for which $\delta=r_{n}/s_{n}$. Then, for $n\gt n_{0}$, it follows that

$\displaystyle s_{n}\left|\delta-\frac{r_{n}}{s_{n}}\right|=\frac{|ps_{n}-qr_{n}|}{q}\ge\frac{1}{q}$.

This contradicts the assumption on $s_{n}$.

### Theorem

$\sqrt{2}$ is irrational.

### Proof

Let $\delta=\sqrt{2}$. Construct two sequences of positive integers by

\begin{align} r_{n+1} &= r_{n}+2s_{n}, \\ s_{n+1} &= r_{n}+s_{n}, \end{align}

starting with $r_{0}=s_{0}=1$. By induction, $r_{n}\ge s_{n}$ and

$2s_{n}^2 = r_{n}^{2}+(-1)^{n}$.

This yields

$\displaystyle \frac{1}{s_{n}^{2}}=\left|2-\frac{r_{n}^2}{s_{n}^2}\right| = \left|\sqrt{2}-\frac{r_{n}}{s_{n}}\right|\cdot \left|\sqrt{2}+\frac{r_{n}}{s_{n}}\right|$.

Therefore,

$\displaystyle s_{n}\left|\sqrt{2}-\frac{r_{n}}{s_{n}}\right|\le s_{n}\left|2-\frac{r_{n}^2}{s_{n}^2}\right|\le \frac{1}{s_{n}}\rightarrow 0.$

By Lemma, $\sqrt{2}\notin\mathbb{Q}$.

### References

1. V. H. Moll, Numbers and Functions: From a Classical-Experimental Mathematician's Point of View, AMS, 2012, p. 45