Why is a union of infinitely many bounded sets not necessarily bounded, please? In addition, what condition can we add to make this union bounded, please?
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3Can you give me an example of a bounded set (say, on the real line)? Can you see why a union of infinitely many of them might not be bounded? – Christopher A. Wong Sep 02 '13 at 09:36
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1$$\bigcup_{n=1}^\infty [-n, n] = (-\infty, +\infty).$$ – njguliyev Sep 02 '13 at 09:37
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$\mathbb{R}^n=\bigcup_{n=1}^{\infty} \{x\in\mathbb{R}^n:\left\|x\right|\leq n\}$
A finite union of bounded sets is bounded.
I hope this helps!
Amitesh Datta
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Each subset of a metric space is a union of bounded sets (the singletons) but not every subset is bounded.
Davide Giraudo
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Since you're asking why it's not necessarily true that such a union is bounded, it suffices to consider a counterexample. Define $A_n = [n, n + 1]$; then
$$\bigcup_{n \in \Bbb{N}} A_n = [0, \infty)$$
is unbounded.
It is necessary and sufficient that there is a common bound on all the $A_n$ for the union of the $A_n$ to be bounded.