Theorem 7.2.5
Claim

Let $A$ be a set. The following statements about $A$ are equivalent.

  1. $A$ is infinite.
  2. $A$ contains a sequence of distinct terms.
  3. $A$ can be put into one-to-one correspondence with a proper subset of itself.
Proof

start typing proof here… Assume that A is an infinite set, that A contains a sequence of distinct terms, and that A can be put into a 1-1 correspondence with proper subset of itself.

$\square$


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