# Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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### prove that the unit circle $x^2+y^2=1$ is a closed set

I need to prove that the unit circle $x^2+y^2=1$ is a closed set in $\mathbb{R}^2$ is closed using convergent sequence method.
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### Is every Cauchy sequence in a non-complete metric space convergent?

A metric space $X$ is called complete if every Cauchy sequence in $X$ has a limit in $X$. For a non-complete metric space $X$, can we say that every Cauchy sequence is convergent? (even though the ...
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### Completeness of the set of convergent sequences

It's a problem from the book "Topology of Metric Spaces", written by Kumaresan: "Show that the set $\textbf{c}$ of convergent sequences in the Normed Linear Space of all bounded real sequences ...
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### Showing that $X= \left\{ u \in C\bigl([0, \infty);E\bigr) \mid \sup_{t \geq 0} e^{-kt} \|u(t)\| < \infty \right\}$ is Banach.

I am working on my degree thesis studying the Hille Yosida Theorem and as a motive to it, I want to present the Cauchy-Lipschitz-Picard Theorem. Haim Brezis highlights the path for a very ...
I have some questions about the following theorem and it's proof. Theorem. Let $X\subset \mathbb R^n$, open, convex, bounded and $f_n:X \to \mathbb R^m$ differentiable.And let's consider the ...