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April 12th, 2012, 02:46 PM   #1
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Every totally bounded set in a metric space is bounded

We discussed this theorem in class, which I zoned out since I was up too late writing a paper and I am totally lost. It seems logical that every totally bounded set in a metric space is bounded but I am have trouble understanding the theorem I randomly wrote down. Can anyone tell me in terms I can understand I would appreciate it.
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April 20th, 2012, 07:23 PM   #2
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Re: Every totally bounded set in a metric space is bounded

Below are the following definitions:

A set is if for each there exist such that .

A set is if there exists and such that .

In the definition of total boundedness, we can make our radius as small as we like, but in the other definition, our radius isn't in under our control. Thus, total boundedness is a stronger condition, and it always implies boundedness. The converse however, it not true in general.
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April 21st, 2012, 02:56 PM   #3
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Re: Every totally bounded set in a metric space is bounded

If you know that what means that is totally bounded I would suggest to choose:
, this means that is bounded.
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