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In one of my math classes, I have a theorem that starts with the following : Let T be a non-empty set, and X a subset of B(T) [that is the set of bounded real value functions] that is closed under addition by positive constant functions, that is f element of X implies f + a is also element of X for any a > 0.

I just don't get how it is possible that I can add any value to a bounded set and stay within the set? There's obviously something I don't understand properly, because adding a big enough constant would cause the value to go off bounds.

Can someone explain me what this actually means? Thanks a lot.

Andrés E. Caicedo
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Daloilpa
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    Each element is a bounded function $\ne$ a bounded set of functions. –  Aug 24 '16 at 22:50
  • For analogy: $\mathbb{N} = {0, 1, 2, 3, ...}$ is a set of numbers, each and every one of which is bounded. But $\mathbb{N}$ is not bounded. –  Aug 24 '16 at 22:52
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    Ahhh, I get it. If you add a finite constant to any bounded function, the function remains bounded. I thought the bounds were on the value the functions could take, but that definition makes more sense. Thanks a lot, really appreciated. – Daloilpa Aug 24 '16 at 22:52

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