(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Working in a banach space [itex](X,\|\cdot\|)[/itex] we have a sequence of compact sets [itex]A_k\subset X[/itex].

Assume that there exist [itex]r_k>0[/itex] such that [itex]\sum_{k\in\mathbb{N}}r_k<\infty[/itex] and for every [itex]k\in\mathbb{N}[/itex]: $$A_{k+1}\subset\{x+u|x\in A_k,u\in X,\|u\|\leq r_k\}.$$Prove that the closure of [itex]\bigcup_{k\in\mathbb{N}}A_k[/itex] is compact.

2. Relevant equations

3. The attempt at a solution

While talking to the teaching assistant it all seemed very doable, but now that I am back home, I am still struggling.

Here is what I was suggested to do:

Since we are in a normed space, then compactness is equivalent to sequential compactness, i.e. existence of a convergent subsequence for every sequence.

Let [itex]\{x_n\}_{n\in\mathbb{N}}[/itex] be a sequence from the closure of [itex]\bigcup_{k\in\mathbb{N}}A_k[/itex]. Then for each n there exists [itex]y_n[/itex] from [itex]\bigcup_{k\in\mathbb{N}}A_k[/itex] such that [itex]\|x_n-y_n\|<\frac{1}{n}[/itex] and it sufficent to show that [itex]\{y_n\}_{n\in\mathbb{N}}[/itex] has a convergent subsequence.

This is the point where I lose my grip and have no idea what to do further

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# Homework Help: Compactness of sets in Banach spaces

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