# Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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### Do Cauchy always converge (for some superset)?

I know Cauchy sequences are only guaranteed converge in complete metric spaces. However, I have been struggling with one issue. It seems that every Cauchy sequence converges, at least in a larger ...
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### Intuition for non-convergence of Cauchy sequence in $\mathbb{Q}$

Suppose we were standing on the rational line at the point 3. Then we took a step to the point 3.1, then to 3.14, etc. (Cauchy sequence of decimal approximations of $\pi$). Suppose, also, that it ...
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### Convergence of complex sequence with gradient descent

I am looking to solve the following otimization: $\underset{{{{\bf x} \in \mathcal{X}}}}{\text{min}} \; f({\bf x})$ where ${\bf x} \in \mathbb{C}^N$ and $f({\bf x}) \in \mathbb{R}$, $f(x)$ is a ...
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### How to prove that $\left\{\frac{1}{n^{2}}\right\}$ is Cauchy sequence
How can I prove that $\left\{\frac{1}{n^{2}}\right\}$ is a Cauchy sequence? A sequence of real numbers $\left\{x_{n}\right\}$ is said to be Cauchy, if for every $\varepsilon>0$, there exists a ...