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Math Help - Proof: finite sets

  1. #1
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    Proof: finite sets

    Let S be a non-empty subset of N. Suppose S has an upperbound (not necessarily in S) b e N. Show that S is finite.

    How would I show this?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by jzellt View Post
    Let S be a non-empty subset of N. Suppose S has an upperbound (not necessarily in S) b e N. Show that S is finite.

    How would I show this?
    clearly S \subseteq B, where B = \{ n \in \mathbb{N} \mid n \le b \}.

    note that B is finite, since it has b (or, depending on your definition of natural numbers, b + 1) elements. since S \subseteq B, S must be finite as well
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    Why is it clear that S is a subset of B?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by jzellt View Post
    Why is it clear that S is a subset of B?
    S has an upper bound. that is, all its elements are less than or equal to b. so what i did was to construct the set of ALL elements of natural numbers that are less than or equal to b. showing S is a subset is not that difficult, for assume x is some element in S, then that means it is a natural number less than or equal to b, but then, that means x is in B, since B is the set of ALL such elements. thus, S is a subset of B.
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