An exercise in the book wants me to show that compactness is a topological property. This chapter is about topology of $R^n$ and we are working with an assumed metric. We haven't come to definition of topological spaces yet. We defined; open sets, closed sets, limit points, sequences, continuous functions, bounded sets, homeomorphisms and topological properties etc.
Here is the definition of compactness that the book uses.
Defn: A set $S$ is compact if every infinite sequence contained in $S$ has a limit point contained in $S$.
Now I am trying to prove the following.
Proposition: Compactness is a topological property.
I am trying to show that if $A \subset R^n$ is compact and there exists a homeomorphism $f: A \to B$, then $B$ is compact.
Strategy: Assume $A$ is compact, and $f$ is continuous with a continuous inverse. Go with proof by contradiction. Suppose $B$ is not compact. Then by Heine-Borel, $B$ is not (closed and bounded). So $B$ is either not closed or not bounded.
I showed that if $B$ is not closed, then $A$ is not compact and found a contradiction. I am trying to also show that if $B$ is not bounded, a contradiction arises. Then I think, I can conclude $B$ must be compact because all roads lead to a contradiction.
I need to show if $B$ is not bounded, a contradiction arises. Possible candidate is to show that $A$ is not compact, which contradicts the hypothesis.