Math and science::Analysis::Tao::05. The real numbers

Reciprocals of real numbers

In order to define reciprocation, it is required to insure that division [...] is avoided. Eventually, we define the reciprocal of a real by taking a Cauchy sequence that represents the real, then taking the reciprocal of each elements to form a new Cauchy sequence. To do this requires the following two lemmas.

Lemma 5.3.14

Let $x$ be a non-zero real number. Then $x = L I M_{n \to \infty} a_{n}$ 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 $x = L I M_{n \to \infty} a_{n}$ , which exists by Lemma [...]. Then we define the reciprocal $x^{- 1}$ as:

x^{- 1} := L I M_{n \to \infty} a_{n}^{- 1}

which we know to be a Cauchy sequence from Lemma [...].

24.08.2019