# Tagged Questions

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### Proving there exist convergent subsequences for bounded sequence of real numbers

I'm trying to teach myself some basic topology by self-studying from Intro to topology by Mendelson. I'm stuck on one of the exercises and can't figure out how to proceed with the proof. The question ...
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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 under ...
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### Prove that $\{n\}$ is a Cauchy sequence that doesn't converge.

Consider the distance function given by $d:\mathbb{R}\times\mathbb{R}\to\mathbb{R},\;d(x,y)=|\int_x^yf(t)dt|$ where $f$ is a continuous and positive function such that $\int_{-\infty}^{+\infty}f$ ...
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### Help undestanding compactness with convergent subsequences

One way to define compactness in metric spaces is to note that in compact metric space each sequence has a convergent subsequence. Understanding compactness is difficult for me from this ...
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### Convergence of the limits of a double sequence in one variable as a sequence of the other variable

If $a_{n,m}$ is a double sequence in a metric space such that $a_{n,m} \rightarrow_m a_n$ uniformly on $n$ and $a_{n,m} \rightarrow_n a$ for all $m$, then $$a_{n} \rightarrow a.$$ Indeed for any ...
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### Prove that the space of divergent sequences in $(l_{\infty},d_{\infty})$ is open and dense. Is it separable?

The problem statements are: Consider the space $A=\{ \{a_n\}_{n \in \mathbb N} \in l_{\infty} : \{a_n\}_{n \in \mathbb N} \text { is not convergent }\}$ $a)$ Prove that $A$ is open and dense in ...