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What are some good examples of sequences which are Cauchy, but do not converge?

I want an example of such a sequence in the metric space $X = \mathbb{Q}$, with $d(x, y) = |x - y|$. And preferably, no use of series.

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12  
Try the sequence $a_k =$ decimal expansion upto k decimal places of $\sqrt{2}$. It converges to $\sqrt{2} \notin \mathbb{Q}$. You should straighten out the details for yourself. –  Ravi Donepudi Mar 12 '12 at 15:43
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Of course, every increasing such sequence can be written as a "series". –  Michael Greinecker Mar 12 '12 at 15:46
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Take any convergent sequence, and throw away the limit. E.g., $x_n=\frac{1}{n}$ on $(0,\infty)$. –  Arturo Magidin Mar 12 '12 at 16:03
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@Michael surely every sequence can be thought of as a series? –  hayd Mar 12 '12 at 17:15
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You are of course correct. –  Michael Greinecker Mar 12 '12 at 17:54

8 Answers 8

up vote 18 down vote accepted

Another one, same idea: $$ a_n = \left(1+\frac{1}{n}\right)^n $$ a sequence of rationals, but its limit $e$ is not rational.

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You take any irrational number, say $\sqrt2$, and you consider its decimal expansion, $$ \sqrt2=1.4142\ldots $$ Then you define $x_1=1$, $x_2=1.4$, $x_3=1.41$, $x_4=1.414$, etc.

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This sort of uses series, though. –  Rasmus Mar 12 '12 at 16:12
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@Rasmus But then every sequence $(a_k)$ can be expressed as partial sums of the series $\displaystyle \sum_{i=0}^\infty (a_{i+1}-a_i)$, so they are really hard to distinguish? :) –  Ravi Donepudi Mar 12 '12 at 16:19

A fairly easy example that does not arise directly from the decimal expansion of an irrational number is given by $$a_n=\frac{F_{n+1}}{F_n}$$ for $n\ge 1$, where $F_n$ is the $n$-th Fibonacci number, defined as usual by $F_0=0$, $F_1=1$, and the recurrence $F_{n+1}=F_n+F_{n-1}$ for $n\ge 1$. It’s well known and not especially hard to prove that $\langle a_n:n\in\Bbb Z^+\rangle\to\varphi$, where $\varphi$ is the so-called golden ratio, $\frac12(1+\sqrt5)$.

Another is given by the following construction. Let $m_0=n_0=1$, and for $k\in\Bbb N$ let $m_{k+1}=m_k+2n_k$ and $n_{k+1}=m_k+n_k$. Then for $k\in\Bbb N$ let $$b_k=\frac{m_k}{n_k}$$ to get the sequence $$\left\langle 1,\frac32,\frac75,\frac{17}{12},\frac{41}{29},\dots\right\rangle\;;$$ it’s a nice exercise to show that this sequence converges to $\sqrt2$.

These are actually instances of a more general source of examples, the sequences of convergents of the continued fraction expansions of irrationals are another nice source of examples; the periodic ones, like this one, are probably easiest.

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If you are not married to using the rationals, I would suggest also using the open interval $(-1,1)$. Here you can take the sequence $( 1 - \frac{1}{n} )_{n=1}^\infty$, and note (quickly) that it is Cauchy and that it should converge to $1$, which of course is not in $(-1,1)$.

The punch line -- if it can be called that -- is that $(-1,1)$ is homeomorphic to the the entire real line $\mathbb{R}$, meaning that they have the same topological structure.

This tells us that it is the underlying metric which tells us whether a sequence is Cauchy or not, and it is not a property of the topology alone. And there are metrics on $(-1,1)$ compatible with the topology in which the aforementioned sequence is not Cauchy; an example would be $$\rho (x,y) = | \tan (\frac{\pi x}{2} ) - \tan (\frac{\pi y}{2}) |.$$

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Here's another, well-known, example: Let $b>0$. Take $a_1>0$ rational and define $a_{n+1}={1\over2}(a_n+{b\over a_n})$. One can show that this sequence is bounded below and eventually monotone decreasing. From this it follows that $(a_n)$ converges to $\sqrt b$.

Taking $b$ to be prime, for example, gives a sequence of rational numbers that converge to an irrational number.

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If we have a subset $A$ of $\mathbb{Q}$ and a limit point $p$ of $A$ such that $p$ is not in $A$, then we can generate such a sequence. Just take any sequence converging to $p$ (which we know exists since $p$ is a limit point of $A$). Furthermore since the sequence is convergent it is also a Cauchy sequence. More precisely for every convergent sequence $p_n$ in the ambient metric space $X$, s.t. $p_n$ is in $A$ for all $n$, we have that $p_n$ is also a cauchy sequence in $A$. Note that the point is that this holds even though $A$ does not contain the limit of the sequence.

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Here is another idea, generalising David Mitra's example. Let $P(x)$ be a polynomial with integer coefficients with an irrational real root $\xi$. Newton's method to find $\xi$ provides a sequence of rationals converging to $\xi$. Take $x_0\in\mathbb{Q}$ close enough to $\xi$. then the sequence defined recursively as $$ x_{n+1}=x_n-\frac{P(x_n)}{P'(x_n)} $$ converges to $\xi$ and $x_n\in\mathbb{Q}$ for all $n$. David's example is obtained taking $P(x)=x^2-b$.

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Such complicated examples! Here's a simple one: $\{1/n\}_{n=1}^\infty$ is a Cauchy sequence in the interval $(0,\infty)$ and does not converge within the interval $(0,\infty)$ (with the usual metric).

Of course you could tack $0$ onto the space and get $[0,\infty)$, and within that larger space it converges. Every metric space has a completion, to a point within which every Cauchy sequence converges.

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