# Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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### Proving convergent sequences are Cauchy sequences

Prove that if $x_n \rightarrow a, n \rightarrow \infty$ then $\{x_n\}$ is a Cauchy sequence. I believe I have found the proof as follows, wondering if there are any simpler methods or added ...
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### Prove the space of bounded sequences is Banach

http://www.math.ucla.edu/~tao/resource/general/121.1.00s/exam1sol.pdf Here is a proof, but I cannot fully understand why it does not give a proof that $x$ is a bounded sequence (i.e. $x$ is in the ...
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### Baby Rudin Exercise 4.13 Alternate Proof Verification

I would like to know if my proof of ex 4.13 is correct. Thanks! Exercise 4.13 in Rudin asks: Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous real function ...
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### Using Cauchy's Criterion to show non-uniform convergence of series of functions.

I want to show $$\sum_{n=0}^\infty x^n$$$$x\in(-1,1)$$ does not converge uniformly using the negation of Cauchy's Criterion for uniform convergence of series of functions. Cauchy's Criterion states ...
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### Proof review - (lack of rigour?) Convergent sequence iff Cauchy without Bolzano-Weierstrass

I am currently trying to improve my skills doing epsilon-delta proves and I just attempted the following one. Since I'm such a newbie in calculus I would like to improve learning form my mistakes (...
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### Showing a sequence is Cauchy; loss of generality?

The exercise is as follows; Show that the sequence $$(a_n) = \left(\frac{(-1)^n}{\sqrt{n}}\right)_{n \in \Bbb N}$$ is a Cauchy Sequence. Solution: Let $m > n.$ Since we are trying to show ...
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### Proving that every Cauchy sequence in measure converges in measure

Let $(X,\mathcal{A},\mu)$ be a measure space and $(f_n)$ a sequence of real-valued functions on $X$ which is Cauchy in measure; that is, for any $\epsilon>0$ there exists $N\in\mathbb{N}$ such that ...
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### Prove that it is a cauchy sequence

Show that the sequence $\langle f_n\rangle$ where $$f_n = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots + \frac{(-1)^{n-1}}{n}$$ is a cauchy sequence. My Approach: I tried solving it by starting it ...
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### Completeness of $l_1 ^\infty$

I'm trying to prove that $l_p ^\infty$ is complete for each $p\geq 1$ but only with the definition of $\varepsilon$-$N$. I know that this have been proved in other posts here but I couldn't find a ...
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### Show that $(3x_{n}+4y_{n})$ is also Cauchy sequence.

Show that if $(x_{n})$ and $(y_{n})$ are Cauchy sequences in $X$, then the sequence $(3x_{n}+4y_{n})$ is also Cauchy sequence using the definition of a Cauchy sequence. Attempt Let $\epsilon > 0$ ...
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### Cauchy sequence that satisfies $\|x_{k+2}-x_{k+1}\|\le\theta\|x_{k+1}-x_k\|$

Suppose the sequence $\{x_k\}_{k=1}^\infty\subset\mathbb{R}^n$ satisfies $\|x_{k+2}-x_{k+1}\|\le\theta\|x_{k+1}-x_k\|$ for all $k\ge1$, with $0<\theta<1$. Show that $\{x_k\}$ is a Cauchy ...
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### Every Cauchy sequence converges

SENTENCE: The p-adic numbers are complete with respect to the p-norm, ie every Cauchy sequence converges. PROOF: Let $(x_i)_{i \in \mathbb{N}}$ a Cauchy-sequence in $\mathbb{Q}_p$. We want to show ...
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### Prove that a norm makes a space Banach

I have to prove that if $A$ is a C*-Algebra then the algebra $A_1$ obtained adjoining the identity is a C*Algebra too (with the usual algebraic operation defined). I have any problem in all the ...
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### Prove that ${\xi_i}$ is complete system in $l_2$

Let ${\{x_i\}} \subset \mathbb{C}$, $x_i \neq 0$, $x_i \rightarrow 0$, $|x_i| < 1$. $\xi_i$ = {$x^k_i$}$_{k\ge0} \Rightarrow$ {$\xi_i$} is complete system of sequences in $l_2$. I should prove ...
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### Patterns appearing in irrational approximation graphs

I'd like to know more about some patterns I found in graphs corresponding to irrational numbers. Here's the graph for $\sqrt 2$ for example First, I'll try to explain most naturally the function that ...
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### What's the diagonal Cauchy subsequence (to show totally bounded if and only if every sequence has a Cauchy subsequence)?

I read a textbook showing a subset of a normed linear space is totally bounded if and only if every sequence in it has a Cauchy subsequence. It proves as follows: $\Rightarrow$ Let $(x_n)$ be an ...
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### Proof that the metric space of convergent sequences is complete

Let $c_{0}$ be the space of real-valued sequences $\{x_{n}\}$ which converge to zero, equipped with the metric $d(\{x_{n}\}, \{y_{n}\}) = \sup\{|x_{n} – y_{n}|: n \in \mathbb{N}\}$. Let $e_{k}$ denote ...
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### Compact subset of the space of all bounded sequences of real numbers

Let $X$ be the metric space of bounded sequences of real numbers $x = (x_n)$ with the metric $d(x,y) = \sup_n |x_n - y_n|$. Show that the set  Y = \{ x = (x_n) \in X \mid |x_n| \leq c_n = \text{...
Is $s_n$ a Cauchy sequence if we only assume that $|s_{n+1} - s_n|\lt \frac{1}{n}$ for all $n\in \Bbb{N}$ My original question was poorly worded, hopefully this make more sense. I get that it is not ...