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.
