Let's analyze the proof you're asking about to understand the concept behind the last sentence.

Overview of the Problem

The second theorem in the notes seems to involve concepts from set theory, particularly related to bijections and infinite sets. The theorem can be roughly stated as:

• Theorem: If $A$ is a set with $n$ elements (finite), and $B$ is an infinite set, then the set $A \cup B$ (union of $A$ and $B$) is also infinite.

Proof Outline

1. Given:

   • $A$ is a finite set with $n$ elements.
   • $B$ is an infinite set.

2. Claim: $A \cup B$ is infinite.

3. Construction of a Function:

   • Define a function $g: B \rightarrow A \cup B$ which maps each element of $B$ to itself in $A \cup B$.
   • If the function $g$ is bijective, $A \cup B$ is infinite.

4. Last Sentence:

   • The proof concludes that $g$ is bijective, implying $A \cup B$ is infinite.

Understanding the Last Sentence

The concept behind the last sentence revolves around the idea that since $B$ is infinite, and $g$ is defined such that it essentially maps elements of (