Definition of Dedekind-infinity: Bijection or Injections? I have been using the following definition of infinity (from Dedekind):
Set $S$ is infinite iff there exists a proper subset $S'$ of $S$ and bijection $f: S\rightarrow S'$.
Occasionally, I have seen "injection" used instead of "bijection." Exactly why is this wrong? Can anyone give me a counterexample demonstrating why these are not equivalent definitions?

Follow-up 8 years later: See my formal proof of the equivalence using a form of natural deduction and basic set theory at: http://dcproof.com/DedekindEquivalentDefinitions.htm
 A: They are equivalent. One direction is clear: a bijection $f$ from $S$ to a proper subset of $S$ is an injection from $S$ to a proper subset of $S$. In the other direction, an injection $f:S\to S'$, where $S'\subsetneqq S$, is a bijection from $S$ to $f[S]\subseteq S'\subsetneqq S$, so $f$ is a bijection from $S$ to a proper subset of $S$.
Note, however, that this is not the usual definition of infinite set. According to the usual definition, an infinite set is one that does not admit a bijection to any finite ordinal, i.e., to any set of the form $\{k\in\Bbb N:k<n\}$. Sets with your property are Dedekind-infinite sets. It is consistent with $\mathsf{ZF}$ that there be infinite sets that are Dedekind-finite, i.e., that do not admit a bijection to any finite ordinal or to any proper subset.
A: The conditions are equivalent (without use of the Axiom of Choice): Suppose $S$ contains a proper subset $S'$ with a bijection $f:S\to S'$. This is in particular an injection.
In the other direction, suppose $S$ contains a proper subset $S'$ with an injection $g:S\to S'$. There is clearly also an injection $h:S'\to S$ (simply the inclusion function) and thus, buy the Cantor-Shroeder-Bernstein Theorem (which does not require the Axiom of Choice), there is a bijection $f:S\to S'$.
Thus, the condition for Dedekind finiteness can be stated in terms of an injection or a bijection from the set to a proper subset. The two are equivalent. 
A: Note that every function is onto its range (not codomain!), so if there is an injection into a proper subset then there is a bijection with a proper subset, which is the range of the injection.
