If $A$ is an infinite set, prove that $A$ has a proper infinite subset

I considered taking this contradictory approach: Let $A_1\in A$. Assume that the proper subset $A\smallsetminus\{A_1\}$ is finite such that there exists a bijection $f:A\smallsetminus\{A_1\} \to C_k$ with $C_k=\{1,2,3,\ldots,k\}$ with $k \in N$.

What I wanted to do next is somehow prove that since $A\smallsetminus\{A_1\}$ is finite, then $A$ is finite, contradicting the given and proving my assumption false. However I have no idea how to proceed from here, or even if what I've done so far is plausible. Any help please?

Solution 1:

Extend the mapping to $C_{k+1}$ by mapping $A_1$ to $k+1$. Then the resulting mapping $A \mapsto C_{k+1}$ is a bijection, and as $C_{k+1}$ has cardinality $k+1$, it follows that $A$ must have cardinality $k+1$ a finite number as well.