# Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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### Slightly alternative proof to the converse part of Cauchy's General Principle

I want to prove that: If $\forall \epsilon >0$, $\exists k \in \mathbb{N}$, such that $| u_{n+p}-u_n| <\epsilon$, whenever $n\geq k$, $p\in \mathbb{N}$, then $\{u_n\}$ is convergent. Proof: [...
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### Direct proof for bounded monotonic sequence is Cauchy

Show that every bounded and monotonic sequence is Cauchy. My proof: Suppose $(a_n)$ is a bounded and monotonic sequence. Since $(a_n)$ is bounded, it follows that $\exists M \in \mathbb{Q}^{+}$ ...
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### Proving a sequence is Cauchy from uniform continuity

Given: $f: X \longrightarrow Y$ is uniformly continuous on $X$, $(x_n)_n \in X$ is a Cauchy sequence. Question: What can you say about the sequence ${f(x_n)}$ ? My attempt: Since $f$ is uniformly ...
Is there an easy example of one and the same space $X$ with two different metrics $d$ and $e$ such that one and the same sequence $\{x_n\}$ is a Cauchy sequence for both metrics, but converges only ...