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Math Help - 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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    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 B is an infinite and B \subseteq A we want to show that A is infinite.
    By the definition \left( {\exists C} \right)\left[ {C \varsubsetneq B} \right] and there is a bijection such that f:B \mapsto C.
    It should be clear that D = \left( {A\backslash B} \right) \cup C is a proper subset of A.

    Define a function g:A \mapsto D as 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 g is a bijection.
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