Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

296 questions
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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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Show, using only the definition, that if $\{X_n\}$ is Cauchy and $C\in\mathbb{R}$ then $\{CX_n\}$ is Cauchy.

Let me know if what I did this correct please. Let $\epsilon>0$ be given, we want to find $N\in\mathbb{N}$ such that $|CX_n-CX_m |<\epsilon$ $\forall n,m\geq N$. $$|CX_n-CX_m |=|C||X_n-X_m |$$ ...
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Prove the sequence $f_{n} = \frac{1}{n^2+1}$ is a Cauchy sequence.

Prove the sequence $f_{n} = \frac{1}{n^2+1}$ is a cauchy sequence. I'm just making sure my logic and reasoning is sound for the above proof: Definition of cauchy sequence: $f_n$ is Cauchy if for all ...
I am given a sequence $(f_n)_n$ where $n\in N$. $f_n : \Re \rightarrow \Re: x \mapsto 1$ $f_1:\Re \rightarrow \Re$ is defined as follows $$f_1 (x) = 1 + \int_0^x f_0 (t) dt$$ One sees that the ...
Prove that $\Bbb Q$ is dense in $\Bbb R$ constructed by Cauchy sequences
Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help! Let $\mathcal{C}$ be the set of Cauchy sequences of rationals. We define an ...