Intereting Posts

Classifying algebraic integers satisfying a positivity condition
Classification of Proper Maps between domains in $\mathbb{R}^n$
Is the set of all $n\times n$ matrices, such that for a fixed matrix B AB=BA, a subspace of the vector space of all $n\times n$ matrices?
What kind of “symmetry” is the symmetric group about?
A high-powered explanation for $\exp U(n)=2\iff n\mid24$?
Why does the ideal $(a+bi)$ have index $a^2+b^2$ in $\mathbb{Z}$?
After swapping the positions of the hour and the minute hand, when will a clock still give a valid time?
Infinite series expansion of $\sin (x)$
How to calculate the inverse of a point with respect to a circle?
Lebesgue measure, do we have $m(x + A) = m(A)$, $m(cA) = |c|m(A)$?
Prove that if function f is monotonic, then it one-to-one
What are the situations, in which any group of order n is abelian
$n$ lines cannot divide a plane region into $x$ regions, finding $x$ for $n$
Max dimension of a subspace of singular $n\times n$ matrices
Eigenvalues for the rank one matrix $uv^T$

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.

- $a_{n+1}=|a_n|-a_{n-1} \implies a_n \; \text{is periodic}$
- Sum of the squares of the reciprocals of the fixed points of the tangent function
- Rearrange the formula for the sum of a geometric series to find the value of its common ratio?
- Convergence of the sum of two infinite series only at $x=\frac12$?
- Find the limit of $\sum \frac{1}{log^n(n)}$
- Prove continuity/discontinuity of the Popcorn Function (Thomae's Function).

- Iterated integral question
- Summing $\frac{1}{a}-\frac{1}{a^4}+\frac{1}{a^9}-\cdots$
- Show that $|\sum _{j=1}^n \frac{\cos (2\pi jx)}{j}| \leq C -\log |\sin (\pi x)| $
- Show that $f(x)=\begin{cases}1&\text{if }x\in\mathbb{Q}\\0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q}\end{cases}$ is discontinuous everywhere
- Find the limit of a recursive sequence
- Limit of $\int_0^1\frac1x B_{2n+1}\left(\left\{\frac1x\right\}\right)dx$
- Evaluate the series $\sum_{n=1}^{\infty} \frac{2^{}+2^{-}}{2^n}$
- Sum the following $\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $
- what is the proof for $\sum_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)} = \frac{1}{4} $
- Finding the Sum of a series $\frac{1}{1!} + \frac{1+2}{2!} +\frac{1+2+3}{3!}+…$

Another one, same idea:

$$

a_n = \left(1+\frac{1}{n}\right)^n

$$

a sequence of rationals, but its limit $e$ is not rational.

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 $$\rho (x,y) = | \tan (\frac{\pi x}{2} ) – \tan (\frac{\pi y}{2}) |.$$

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.

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, within which every Cauchy sequence converges.

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.

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.

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 $P(x)=x^2-b$.

It is pretty simple to see an example using the following:

1.) Take any sequence of points that converges to a limit, which is not one of the terms in the sequence.

2.) Delete the limit from the metric space

3.) We have a Cauchy Sequence which is not convergent.

For Example:

The sequence 1, 1/2, 1/3, 1/4, 1/5, …

We know that this converges to 0. So, now take the Metric Space R (all real numbers) and delete the limit 0 from the metric space.

So the Cauchy sequence 1, 1/2, 1/3, 1/4, 1/5, … is not convergent in the metric space R – {0}.

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.

How about the Cauchy sequence $$x_1=0.1$$ $$x_2=0.12$$ $$x_3=0.123$$ $$x_4=0.1234$$ $$x_5=0.12345$$ $$x_6=0.123456$$ $$\cdots$$in $\mathbb{Q}.$

This agood example for acauhy seqeance

but is not convergent.Take Xn€Q suhc that

Xn^2<2 this imples Xn<2^2

then Xn convergent to 2^2 which dont belong to Q.

So is not convergent.

- Show that a positive operator on a complex Hilbert space is self-adjoint
- Calculating Bernoulli Numbers from $\sum_{n=0}^\infty\frac{B_nx^n}{n!}=\frac x{e^x-1}$
- Inequality involving the sup of a function and its first and second derivatives
- The Conjecture That There Is Always a Prime Between $n$ and $n+C\log^2n$
- central limit theorem for a product
- Primes and probabilities
- A sequence of real numbers such that $\lim_{n\to+\infty}|x_n-x_{n+1}|=0$ but it is not Cauchy
- I need help with Differential equations.
- Find the general solution of the differential equation $\left(3y^2+x^2+x+2y+1\right)\cdot y'+2xy+y=0$
- Geometric Proof, Tournament of Towns Fall 2015 Junior A-Level
- Using Zorn's lemma to show that every field has an algebraic closure.
- Inequality using Cauchy-Schwarz
- Numerically Efficient Approximation of cos(s)
- Reference for Topological Groups
- Characterizing units in polynomial rings