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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 by zero 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 $$x = LIM_{n\rightarrow \infty}a_n$$ 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:

$x^{-1} := LIM_{n\rightarrow \infty}a_n^{-1}$

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

Tao, Analysis I