nested_quantifiers_quiz.tex

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\activitytitle{Quiz on nested quantifiers and negation of quantifiers}{40 points}

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\item Show that for all real numbers $a > 0$, there exists an integer $n$ such that for all $m > n$, $\frac{1}{\sqrt{m}} < a$.
Since there are three quantifiers, you will use the word ``let'' three times.
Be clear what is happening each time!

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\item Rewrite the statement in Problem 1 using quantifiers.

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Now, negate the statement and push the ``not'' past all of the quantifiers.

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\item On the back of the page, show that for all real numbers $x$ and $y$ with $0 < x < y$, we have $x^2 < y^2$.
It's not enough to say that the result is obvious or just look at examples, you need to find a way to break this into smaller steps that you can verify.
Make note of each time you use one of the assumptions in the proof.

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Extra credit for writing two distinct proofs.
For example, for a second proof, you could use integrals.
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