Prove that $f^{-1}$ is continuous

Question

Let $M$ be a compact metric space ($N$ is a metric space too) and let $f:M \to N$ be a continuous bijection. Prove that $f^{-1}$ is continuous.

My proof.

Let $A \subset M$ be closed. Then $A$ is compact. It follows that $f(A)$ is compact. Then since every compact set is closed and bounded, we conclude that $f(A)$ is closed and hence $f^{-1}$ is continuous.

I noticed that my proof didn't use the fact that $f$ is a bijection, so where did all these go horribly wrong? I would appreciate if someone guided me along! Thank you!

Answer 1

You use fact that you have a bijection for

$$f^{-1}\left(f(A)\right)=A=$$

Of course, you are also using that $\;\left(f^{-1}\right)^{-1}=f\;$...