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Prove that a set A is infinite if and only if $A$ contains a proper subset $B$ that satisfies $|B|=|A|$.

For the first part, I tried the following:

Since $A$ is infinite, it has a countable subset $S=\{a_n\}$ with $n$ in $\mathbb N$. Then we have a function $f:A\to A\setminus \{a_0\}$ such that $f(a_n) = a_{n+1}$ if $a$ is in $A$ and $a=a_n$ and $f(a)=a$ otherwise.

I don’t know if this approach is correct, and need help proving part 2.

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That approach is excellent: you have indeed shown that if $A$ is infinite, it admits a bijection with a proper subset.

Suppose that $f:A\to A$ is a bijection from $A$ to a proper subset of $A$. Pick $a_0\in A\setminus f[A]$. Given $a_n$, let $a_{n+1}=f(a_n)$; now prove that the map $n\mapsto a_n$ is a bijection from $\Bbb N$ into $A$.

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