Can You Think of a Cauchy Sequence Which Is Not Convergent?

This commodity/section deals with mathematical concepts appropriate for a educatee in mid to late high schoolhouse.

The reader should be familiar with the material in the Limit (mathematics) page.

A Cauchy sequence (pronounced KOH-she) is an infinite sequence that converges in a particular style. This type of convergence has a far-reaching significance in mathematics. Cauchy sequences are named afterwards the French mathematician Augustin Cauchy (1789-1857).

In that location is an extremely profound aspect of convergent sequences. A sequence of numbers in some fix might converge to a number not in that set. The famous case of this is that a sequence of rationals might converge, but not to a rational number. For example, the sequence

1.4

1.41

1.414

i.4142

1.41421

consists merely of rational numbers, but it converges to , which is not a rational number. (Meet real number for an outline of the proof of this.)

The sequence given above was created by a computer, and information technology could be argued that we haven't really exhibited the sequence. But we tin can put such a sequence on a firm theoretical basis past using the Newton-Raphson iteration. This would give us

so that

...

These aren't the aforementioned every bit the sequence given previously, only they are all rational numbers, and they converge to .

So if we lived in a earth in which nosotros knew about rational numbers only had never heard of the real numbers (the aboriginal Greeks sort of had this trouble) nosotros wouldn't know what to do nigh this. Recall that, for a sequence (a_n) to converge to a number A, that is

we would need to apply the definition of a limit—we would need a number A such that, for every ε > 0, in that location is an integer M such that, whenever .

There is no such rational number A.

But there is clearly a sense in which converge. The definition of Cauchy convergence is this:

A sequence converges in the sense of Cauchy (or is a Cauchy sequence) if, for every ε > 0, there is an integer One thousand such that any two sequence elements that are both across One thousand are within ε of each other.

Whenever and , .

Note that in that location is no reference to the mysterious number A—the convergence is defined purely in terms of the sequence elements being close to each other. The example sequence given in a higher place can be shown to be a Cauchy sequence.

Construction of the Real Numbers

What we did to a higher place effectively defined in terms of the rationals, past proverb

"The square root of 2 is whatever the Cauchy sequence given above converges to."

even though that isn't a "number" according to our limited (rationals-but) understanding of what a number is.

The real numbers can be defined this way, by maxim that a real number is divers to be a Cauchy sequence of rational numbers.

There are many details that we won't piece of work out hither; amongst them are:

• There are different Cauchy sequences that converge to the aforementioned thing; we gave ii sequences above that converged to . And then a real number is actually an "equivalence class" of Cauchy sequences, under a carefully defined equivalence. This is a bit tricky.
• We take to show how to add, subtract, multiply, and divide Cauchy sequences. This is a bit tricky.
• We have to give the Cauchy sequences corresponding to rational numbers. This is easy—5/12 becomes (5/12, v/12, 5/12, ...).

Once we have done that, the payoff is enormous. Nosotros have defined an extension to the rationals that is metrically complete—that extension of the rationals is the real numbers. Metrically consummate means that every Cauchy sequence made from the prepare converges to an element which is itself in the set. The reals are the metric completion of the rationals.

The utilize of Cauchy sequences is i of the 2 famous ways of defining the real numbers, that is, completing the rationals. The other method is Dedekind cuts

External links

• "Cauchy Sequence" From MathWorld
• "Completion" From MathWorld

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Source: https://www.conservapedia.com/Cauchy_sequence

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