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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).

José Carlos Santos
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Alex
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1 Answers1

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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?

diracdeltafunk
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  • Negation of "$f(x_0) < \infty$ for some $x_0 > 0$" should be "...for ALL $x_0>0$"? – Alex May 21 '20 at 20:54
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    Yup! Hopefully it makes intuitive sense why this is the case, but if you just want to remember this as a formal rule for negating quantified statements that's fine too. – diracdeltafunk May 21 '20 at 20:56
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    No, that makes total sense (but the proof much harder :O). – Alex May 21 '20 at 20:57
  • I actually do think proving the contrapositive is a good way to go here, but certainly it'll be harder than you originally thought ;) – diracdeltafunk May 21 '20 at 21:00