# Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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### prove that the sequence $g_n(x)$ is uniformly Cauchy

Let $g : \mathbb{R} \rightarrow \mathbb{R}$ be uniformly continuous and for each $n \in \mathbb{N}$ let $g_{n}:\mathbb{R} \rightarrow \mathbb{R}$, $x \mapsto g_{n}(x) = g \left( x + \frac{1}{n}\right)$...
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### Is this a valid way of proving this series converges?

Take the series $\sum_{k=1}^\infty \frac{(-1)^k}{2^k+k}$ Since I use the Cauchy Criterion, for $n, m$ ∈ ℕ with $m > n$, define: $S_{m,n} = \sum_{k=n+1}^m \frac{(-1)^k}{2^k+k}$ Here is my (shortened)...
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### the description of Cauchy principle of convergence

Question：Is description $$\forall\varepsilon\gt0,\exists N\in\mathbb{Z^+},\text{s.t.}\forall n\gt N,\left| a_n-a_N\right|\lt\varepsilon$$ equivalent to convergence of the suquence $\{a_n\}$? Attempt: ...
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1 vote
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### Problem on cauchy sequences

$\left\{\frac{1}{n}\right\}$ is a Cauchy sequence. Determine $N_0$ such that $|u_n - u_m| < 0.021$, whenever $m, n > N_0$. a. $48$ b. $45$ c. $46$ d. $47$ The correct answer for this was $48$. ...
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