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

1. 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.

2. Originally Posted by noles2188
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?

3. 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."

4. 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.