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Let F be an ordered field with least upper bound property.

1.Let $\alpha: \mathbb{N} \to F$ be a Cauchy sequence. Since F is an ordered field, $x$ is bounded both above and below.
2.By assumption and dual of it, $A$={$\alpha(n)$|$n\in \mathbb{N}$} has a inf $a_0$ and sup $b_0$.
3.F is Archimedean
4.If subsequence of a Cauchy sequence is convergent to $a\in F$ then the Cauchy sequence is convergent to $a\in F$

These are all i know.. How do I prove all Cauchy sequences are convergent in $F$?

Please consider my level. I want quite a direct proof not mentioning any topology & Cauchy net.

*Comment button is not available to me now, (I don't know why), so i write this here. I just proved it with facts that (i)every cauchy sequence is convergent in the set of Cauchy reals and (ii)there exists a bijective homomorphism between two dedekind complete fields and (iii)the set of Cauchy reals is dedekind complete. Let $x:i→x(i):\mathbb{N}→F$ be a cauchy sequence in dedekind complete field $F$. Then use the bijective homomorphism $f$ to show that $x':i→f(x(i))$ is a cauchy sequence in the set of Cauchy reals. By the fact (i), $x'$ is convergent. Since inverse of $f$ is also homomorphism, use this to show that $x$ is convergent.

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2 Answers

up vote 1 down vote accepted

Let $c_0 = \frac{a_0 + b_0}2$ (Note that an ordered field has characteristic 0, so $2 = 1+1$ is invertible in $F$). If $[a_0, c_0]$ contains infinitely many $\alpha(n)$, then let $a_1 = a_0$, $b_1 = c_0$, otherwise $a_1 = c_0$, $b_1 = b_0$. Now set $c_1 = \frac{a_1 + b_1}2$. Continuing inductively, we obtain sequences $(a_n)$, $(b_n)$ such that:

  • $[a_n, b_n]$ contains infinitely many $\alpha(k)$.
  • $b_n - a_n = \frac{b_0 - a_0}{2^n}$
  • $a_{n} \le a_{n+1} \le b_{n+1} \le b_n$.

hold for each $n$.

Now let $a^* = \sup_n a_n$ (note that $a_n \le b_0$, so the sup exists) and $b_* = \inf b_n$. Then $a_n \le a^* \le b_* \le b_n$ for each $n$, so $b_* - a^* \le b_n - a_n = \frac{b_0 - a_0}{2^n}$. As $F$ is Archimedian, $b_* - a^* = 0$, i. e. $b_* = a^*$.

Set $k_0 := 1$, and for each $n$, given $k_n$ choose $k_{n+1} > k_n$ with $\alpha(k_{n+1}) \in [a_n, b_n]$ (which is possible since the latter interval contains infinitely many $\alpha(k)$).

We will prove, that $\alpha(k_n) \to a_*$ (which, by 4., suffices). So let $\epsilon > 0$, as $F$ is Archimedian, there is some $N$ with $2^{-N} \le \epsilon(b_0 - a_0)$. For $n \ge N$ we now have $\alpha(k_n) \in [a_N, b_N]$ and $a^* \in [a_N, b_N]$, hence $|a_n - a^*| \le b_N - a_N \le 2^{-N}(b_0 - a_0) \le \epsilon$.

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First of all this is not really a field theory question, but rather an honors calculus / basic real analysis question. Essentially you want a proof that a Cauchy sequence of real numbers is convergent, in particular a proof which relies on the least upper bound axiom (Dedekind completeness).

A (rather standard) approach to this can be found for instance in Chapter 10 of these notes. The main ingredient that you seem to be missing is the Bolzano-Weierstrass Theorem: in a Dedekind complete ordered field, every bounded infinite sequence admits a convergent subsequence. Do you see how this combines with what you stated that you already know to solve the problem?

The aforelinked notes contain (of course) a proof of Bolzano-Weierstrass. Following the way I was taught this material some years ago, I favor the approach via the Rising Sun Lemma: any infinite sequence (in any linearly ordered set!) admits a monotone subsequence. Thus you reduce to the fact that in any Dedekind complete ordered field any bounded monotone sequence converges (to its supremum / infimum).

Since you are thinking in terms of ordered fields, you might be interested to know that in an ordered field $F$, the following are equivalent:
(i) (Dedekind completeness) Every nonempty bounded above subset has a least upper bound.
(ii) Every bounded increasing sequence converges to its supremum.
(iii) (Bolzano-Weierstrass) Every bounded sequence admits a convergent subsequence.

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