# Reciprocals of real numbers

#### Lemma 5.3.14

Let \( x \) be a non-zero real number. Then \( x = LIM_{n\rightarrow \infty}a_n \) for some Cauchy sequence \( (a_n)_{n=1}^{\infty} \) which is bounded away from zero.

In other words, out of the many Cauchy sequences that can represent the real \( x \), there must be at least one that is bounded away from zero.

The proof for Lemma 5.3.14 is a good exercise and is listed out in the book.

#### Lemma 5.3.15

Suppose that \( (a_n)_{n=1}^{\infty} \) is a Cauchy sequence which is bounded away from zero. Then the sequence \( (a_n^{-1})_{n=1}^{\infty} \) is also a Cauchy sequence.

Finally we get to the definition:

### Reciprocation for reals

Let \( x \) be a non-zero real number. Let \( (a_n)_{n=1}^{\infty} \) be a Cauchy sequence bounded away from zero such that \( x = LIM_{n\rightarrow \infty} a_n \), which exists by Lemma 5.3.14. Then we define the reciprocal \( x^{-1} \) as:

which we know to be a Cauchy sequence from Lemma 5.3.15.