# Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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### Adjacent vs. All-Possible Distances in the Definition of a Cauchy Sequence

I'm currently learning about Cauchy sequences, and I'm trying to build my intuition regarding its definition. My question is on how we can motivate why we consider the distances between all points and ...
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### Cauchy but not convergent sequence on $C[-1,1]$

Consider $C[-1,1] = \{ f: [-1,1] \rightarrow \Bbb{R}: f$ is continuous $\}$, the inner product $\langle f,g \rangle = \int_{-1}^1 f(x)g(x) \, dx$ and $\lVert \cdot \rVert$ the induced norm. Then, the ...
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### Unable to come up with correct bounds for showing sequence convergence. Analysis I, Terence Tao, Theorem 6.1.19, part (g)

I was trying to prove the following (part (g) of the Theorem): Theorem 6.1.19 (Limit Laws). Let $(a_n)_{n=m}^{\infty}$ and $(b_n)_{n=m}^{\infty}$ be convergent sequences of real numbers, and let $x,y$...
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### Need help with proof of an auxiliary result in Analysis I, Terence Tao

I attempt to prove the below result which is needed to prove an overall result in Analysis I, Terrance Tao. For reference, that overall result is Theorem 6.1.19 (Limit Laws) part (e). Any sequence ...
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### Does continuity on a closed set preserves Cauchy sequences?

Let $f$ be a real-valued continuous function on $D=[0,\infty)$. Does $f$ preserve Cauchy sequences, i.e., if $\{x_n\}$ is Cauchy, is $\{f(x_n)\}$ Cauchy? Since $D$ is closed, the Cauchy sequence has a ...
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### Given a sequence $a_n$ s.t. forall $\varepsilon>0$ exists $m,n_{0} \in \mathbb{N}$ s.t. $\forall n \geq n_0$ $|a_n-a_m|<\varepsilon$

So I have that question: Suppose a sequence $a_n$ satisfies that forall $\varepsilon>0$ exists $m,n_{0} \in \mathbb{N}$ s.t. $\forall n \geq n_0$ $|a_n-a_m|<\varepsilon$ determine if that ...
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### Not decreasing sequence that converges to 0: sufficient condition for the sequence of indexes for which the sequence decreases to be bounded.

Consider a strictly positive sequence $\{u_n\}$ of real numbers (i.e. $\forall n, u_n >0$). Suppose that $\displaystyle \lim_{n \to \infty} u_n = 0$ and that $\{u_n\}$ is not decreasing. Consider ...
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### Proof to show Sequence is Cauchy correct?

Given the two facts The sequence $\{ x_n \}$ is Cauchy in the $L^2(D)$ norm. For each $x$ we can apply the following inequality $\|x\|_{E(D)} \leq k \|x\|_{L^2(D)}$ where $k > 0$ I want to show ...
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