# Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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### Cauchy sequence to prove Integral does not exist

I'm learning Cauchy sequence right now and didn't succeed. prove that: $$\int_{i=1}^\infty \cfrac{x\,dx}{\sqrt{1+3x+x^2}}$$ Does not exist Using only Cauchy sequence! Thx!
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### Prove that ln(n) is not a Cauchy Sequence? [duplicate]

Show from the definition of a Cauchy Sequence that Xn=ln(n) is not a Cauchy Series
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### Cauchy sequence is bounded? (Do we need any element in a sequence to be finite?)

This question is related to the question Is every cauchy sequence bounded? The sequence $\{a_n\}$ used in that question $$a_n=\frac{1}{n-1}$$ has the first element $a_1\rightarrow\infty$...
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### Show that $f$ can be extended to a Cauchy-sequence preserving continuous mapping on $\overline{A}$.

Question: Let $(X,d)$ be a metric space and $A\subset X.$ If $f: A\to\mathbb R$ be a Cauchy-sequence preserving continuous mapping then show that $f$ can be extended to a Cauchy-sequence preserving ...
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### Prove Cauchy criterion without using the bounded property

The Cauchy criterion states that: A sequence is Cauchy if and only if it converges. Almost all proofs I can find for this theorem use the fact that: ...
I am currently working on a proof and I have the current properties: A Cauchy sequence $\{ f_n\}_{n\in \mathbb{N}}$ of functions in some space $X$ with the property that $|(f_n - f_m)(x)|<\... 2answers 79 views ### How do I show that if$p_n q_{n-1} - p_{n-1}q_n = 1$, then$p_n/q_n$converges? Let$p_{n}$and$q_{n}$be strictly increasing, integer valued, sequences. Show that if $$p_{n}q_{n-1}-p_{n-1}q_{n}=1,$$ for each integer$n \ge 1$, then the sequence of quotients$\frac{p_{n}}{q_{n}}$... 0answers 51 views ### Show that every Cauchy sequence which admits a convergent subsequence is also convergent. Is my proof correct? Let$(x_{n})_{n=m}^{\infty}$be a Cauchy sequence in$(X,d)$. Suppose that there is some subsequence$(x_{f(n)})_{n=m}^{\infty}$of this sequence which converges to a limit$x_{0}\in X$. Then the ... 2answers 35 views ### Basic Cauchy sequence convergence question Let$f: ]0,1[ \to \mathbb{R}$be a function. Suppose that for every sequence$(\epsilon_n)_n$in$]0,1[$with$\epsilon_n \searrow 0$we have that$(f(\epsilon_n))_n$is a Cauchy sequence. Can we ... 0answers 23 views ### Proving that Cauchy Summation converges [duplicate] How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1)\times (n-k+1)}$$ I tried to prove that by proving that the following summation in in ... 2answers 79 views ### Cauchy product summation converges I had a previous question here, which I'm quoting: How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1) (n-k+1)}$$ I tried to prove ... 0answers 67 views ### Prove that absolute convergence of sup norms implies uniform convergence of functions (Weierstrass M-test) Let$(X,d)$be a metric space, and let$f_{n}$be a sequence of bounded real-valued continuous functions on$X$such that the series$\sum_{n=1}^{\infty}\|f_{n}\|$is convergent. Then the series$\...
We have a Cauchy sequence in the normed space of bounded functions $(f_n)_n \subset B(\Omega, \mathbb{K})$. I have shown that $(f_n(\omega))_n$ is a Cauchy squence for all $\omega \in \Omega$. What's ...