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Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

A sequence $\{x_n\}$ in an arbitrary metric space, and in particular the space $\Bbb{R}$, is called Cauchy if the terms of the sequence become arbitrarily close together; that is, for every $\epsilon > 0$, there exists an $N$ such that

$$n, m \ge N \implies d(x_n, x_m) < \epsilon$$

where $d$ is the distance function for the metric space. In the particular case of the real numbers, this condition becomes

$$n, m \ge N \implies |x_n - x_m| < \epsilon$$

A complete metric space is a metric space in which every Cauchy sequence is convergent; this gives an alternate definition of convergence of a sequence that does not rely on the limiting value.

Source: the Cauchy sequence article on Wikipedia.

2503 questions
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Does every Cauchy sequence converge to *something*, just possibly in a different space?

Question. If I attempt to prove that space $X$ is complete by pursuing the strategy, “Assume $x_n \rightarrow x$; the space $X$ is complete if $x \in X$,” then why is that wrong? Context. I know the definition of Cauchy sequences and convergent…
1Teaches2Learn
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Completion of rational numbers via Cauchy sequences

Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
Derek Scavo
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Why do we want complete spaces? We don't we just use closed spaces?

Why do we care about the notion of a space being complete? Why don't just consider closed spaces? If the space is closed we know that the limits of a sequence exist and are in the set which is a property that is obviously desirable. So what is the…
csss
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Cauchy sequence is convergent iff it has a convergent subsequence

Prove that if $\left ( x_{n} \right )$ is a Cauchy sequence in a metric space X then $\left ( x_{n} \right )$ is convergent if and only if $\left ( x_{n} \right )$ has a convergent subsequence. Note: I realize that this question has been asked…
Dome
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Prove that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ converges absolutely

I'm trying to re-learn my undergrad math, and I'm using Stephen Abbot's Understanding Analysis. In section 2.7, he has the following exercise: Exercise 2.7.5 (a) Show that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ also converges…
Richard Sullivan
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What is the difference between Cauchy and convergent sequence?

I am really confused. I will appreciate if somebody can help me to define the difference between Cauchy and convergent sequence. Many thanks.
learner
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If a subsequence of a Cauchy sequence converges, then the whole sequence converges.

Let $(X,d)$ be a metric space, and say $(x_n)$ is a Cauchy sequence such that it has a convergent subsequence $(x_{n_k})$ that converges to $x$. We show $x_n \to x$. Let $\epsilon > 0$. Take $N >0$ such that for all $n,m > N$, we have $$d(x_n,x_m)…
user124140
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A "non-trivial" example of a Cauchy sequence that does not converge?

A Cauchy sequence doesn't necessarily converge, e.g. take the sequence $(1/n)$ in the space $(0,1)$. Maybe my intuition is wrong but I tend to think of this as, "it does converge but what it converges to is not in the space". Are there any…
user41728
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Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?

What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy? I know that Cauchy condition means that for each $\varepsilon>0$ there exists $N$ such that $d(x_p,x_q)<\varepsilon$ whenever $p,q>N$.
Ashley
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Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$ be a function and suppose for any Cauchy sequence $(a_n)$ in $X$, $(f(a_n))$ is a Cauchy sequence in $Y$. Is $f$ continuous? Let $f$ be continuous, is it uniformly continuous?
user59671
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How do I prove a uniformly continuous function preserves Cauchy sequences?

Let $f$ be a uniformly continuous function on A of $\Bbb{R}$. How do I show that if $a_n$ is Cauchy, then $f(a_n)$ is Cauchy. This is what I have worked on, but it does not quite make sense since I feel like I didn't really use the given condition…
Akaichan
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Proving that a sequence such that $|a_{n+1} - a_n| \le 2^{-n}$ is Cauchy

Suppose the terms of the sequence of real numbers $\{a_n\}$ satisfy $|a_{n+1} - a_n| \le 2^{-n}$ for all $n$. Prove that $\{a_n\}$ is Cauchy. My Work So by the definition of a Cauchy sequence, for all $\varepsilon > 0$ $\exists N$ so that for $n,m…
Moderat
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Recurrence relations and limits, tough.

I would like a hint for the following, more specifically, what strategy or approach should I take to prove the following? Problem: Let $P \geq 2$ be an integer. Define the recurrence $$p_n = p_{n-1} + \left\lfloor \frac{p_{n-4}}{2}…
Areyouheaty
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Why does $ a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$ converge to an irrational number?

There is a problem in my textbook that goes like this $$ a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$$ and $$a_0 =1$$ for all $n\ge1$. It is monotonically decreasing sequence of rational numbers and bounded below. However, it cannot converge to…
Black Jack 21
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Prove that a Cauchy sequence is convergent

I need help understanding this proof that a Cauchy sequence is convergent. Let $(a_n)_n$ be a Cauchy sequence. Let's prove that $(a_n)_n$ is bounded. In the definition of Cauchy sequence: $$(\forall \varepsilon>0) (\exists n_\varepsilon\in\Bbb…
lmc
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