Here's Corollary 6.6 in Topology by James R. Munkres, 2nd edition:
If $B$ is a subset of the finite set $A$, then $B$ is finite. If $B$ is a proper subset of $A$, then the cardinality of $B$ is less than the cardinality of $A$.
How do I supply a proof of this corollary using the preceding material in section 6 of Munkres?
Here are the results that have gone before Corollary 6.6.
Lemma 6.1: Let $n$ be a positive integer. Let $A$ be a set; let $a_0$ be an element of $A$. There exists a bijective correspondence $f$ of the set $A$ with the set $\{1, \ldots, n+1\}$ if and only if there exists a bijective correspondence $g$ of the set $A - \{a_0\}$ with the set $\{1, \ldots, n\}$.
Theorem 6.2: Let $A$ be a set; suppose that there exists a bijection $f \colon A \to \{1, \ldots, n \}$ for some $n \in \mathbb{Z}_+$. Let $B$ be a proper subset of $A$. Then there exists no bijection $g \colon B \to \{ 1, \ldots, n\}$; but (provided $B \neq \emptyset$) there does exist a bijection $h \colon B \to \{ 1, \ldots, m \}$ for some $m < n$.
Corollary 6.3: If $A$ is finite, there is no bijection of $A$ with a proper subset of itself.
Corollary 6.4: $\mathbb{Z}_+$ is not finite.
Corollary 6.5: The cardinality of a finite set $A$ is uniquely determined by $A$.
Although I've understood the above results quite thoroughly (along with their proofs!), I'm unable to figure out how to come up with a proof of Corollary 6.6 using the preceding material in the section.
This is a direct corollary of Theorem 6.2. If $A$ is finite, what can we conclude ? As $B$ is a subset of $A$, what can we conclude ?