I struggle a bit with contraposition and would like to know if my approach is right(is my contraposition statement right)
I have to show the following statement:
If $f(x_0) < \infty$ for some $x_0 > 0$, then $f(x) < \infty$ for all $x > 0$. f is an increasing and concave function.
My idea to prove this was to use contraposition, that means for me (please correct me if I am wrong here) if I can show:
If $f(x) \geq \infty$ for all $x > 0$ (that what would be my assumption in the contraposition proof), then $f(x_0) \geq \infty$ for some $x_0 > 0$.
(That seems too easy to me, because the last inference seems obvious but the initial statement is rather difficult for me to comprehend, so I guess there must be some mistake in the reasoning).
Indeed your statement of the contrapositive is not correct. For example, you thought that the negation of "$f(x) < \infty$ for all $x > 0$" was "$f(x) \geq \infty$ for all $x > 0$", but instead it should be "$f(x) \geq \infty$ for some $x > 0$".
Please try to think this through until it makes sense to you why the negation should be like this! In words: if the statement "$f(x) < \infty$ for all $x > 0$" is false, then it is not the case that $f(x) < \infty$ for all $x > 0$, so it must be the case that $f(x)$ is not less than $\infty$ for at least one value of $x > 0$.
Likewise, you negated the statement "$f(x_0) < \infty$ for some $x_0 > 0$" incorrectly as well. Can you figure out what the negation should really be?