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Thread: Prove: every set with an infinite subset is infinite

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    Prove: every set with an infinite subset is infinite

    Prove that every set that has an infinite subset is infinite.
    Also, prove that every subset of a finite set is finite.
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  2. #2
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    Quote Originally Posted by noles2188 View Post
    Prove that every set that has an infinite subset is infinite.
    Also, prove that every subset of a finite set is finite.
    The proof completely depends on how you textbook defines infinite.
    What definition does it use?
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    Here is the definition from the text: "The statement that the set A is infinite means that there is a nonempty proper subset B of A such that there is a one-to-one correspondence between A and B; A is finite means that A is not infinite."
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    Suppose that $\displaystyle B$ is an infinite and $\displaystyle B \subseteq A$ we want to show that $\displaystyle A$ is infinite.
    By the definition $\displaystyle \left( {\exists C} \right)\left[ {C \varsubsetneq B} \right]$ and there is a bijection such that $\displaystyle f:B \mapsto C$.
    It should be clear that $\displaystyle D = \left( {A\backslash B} \right) \cup C$ is a proper subset of $\displaystyle A$.

    Define a function $\displaystyle g:A \mapsto D$ as $\displaystyle g(x) = \left\{ {\begin{array}{rl} {f(x),} & {x \in B} \\ {x,} & {x \in A\backslash B} \\ \end{array} } \right. $
    Now your task is to show that $\displaystyle g$ is a bijection.
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