1. Part 1 should start as: let $y\in f(f^{-1}(H))$. Then $y=f(x)$ for some $x\in f^{-1}(H)$ ...

Part 2 should start as: let $y\in H$. Now, by surjectivity of $f$, there exists an $x\in A$ such that $f(x)=y$ ...

2. 1)

Let $y\in f(f^{-1}(H))$. Then $y=f(x)$ for some $x\in f^{-1}(H)$.
$f^{-1}(H)=\{x\in A: f(x)\in H\}$,
hence if $x\in f^{-1}(H)$ then $y=f(x)\in H$.

2)

Let $y\in H$. Now, by surjectivity of $f$, there exists an $x\in A$ such that $f(x)=y$.
Since $f(x)=y$ and $y\in H$ we have that $x \in f^{-1}(H)$ and so $f(x)=y \in f(f^{-1}(H))$.

Not too pleased with the last one..

Thanks!

3. Originally Posted by ojones
Plato:

"Proofs by contradiction should not be used if a direct proof is available" - Paul Halmos on mathematical writing.

"All students are enjoined in the strongest possible terms to eschew proofs by contradiction!" - Halsey Royden in Real Analysis.
By the mathematicians we picked I guess it's Topology vs. Measure Theory :P

4. Not really Drexel28. I was just trying to find comments by known mathematicians on the subject. RL Moore is more famous for teaching than research and his comments should be taken in that context.

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