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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 [...] 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 [...]. 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 [...].