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Let $C_b((a,b))$ be the space of continuous and bounded functions on $(a,b) \subset \mathbb{R}$, let $||f||_\infty := \sup\{|f(x)| : x \in (a, b)\}$. Is ($C_b((a,b)),||\cdot||_\infty)$ a complete metric space?

I really have no idea how to proceed with this problem...

Sorry for my english skills, english is not my first language...

ocsecnarf ittorettul
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  • What does it mean to be a complete metric space? – fwd Apr 21 '21 at 17:51
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    Every cauchy sequence is convergent –  Apr 21 '21 at 17:51
  • It is complete if every Cauchy sequence converges. So to prove it is complete, you take an arbitrary Cauchy sequence and show it has a limit, IE a function in the space it approaches. To show it isn't complete, you need to find one cauchy sequence that doesn't converge – Alan Apr 21 '21 at 17:51
  • I think is is not a complete metric space, but i cant find any example for it... –  Apr 21 '21 at 17:54
  • The space you are loking for is $C((a,b))$ or $C([a,b])$? – ocsecnarf ittorettul Apr 21 '21 at 17:56
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    It is actually C((a,b)) –  Apr 21 '21 at 18:00
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    The problem is that on $C((a,b))$, $||\cdot||_\infty$ in not a well defined norm, since in could be infinite. You must restrict to the space $C_b(a,b)$ i.e. the space of continuous and bounded function on $(a,b)$, then the answer is affermative, it is a Banach space – ocsecnarf ittorettul Apr 21 '21 at 18:19

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