From: Stephen Dalton 
Date: November 20, 2009 7:53:56 PM EST
To: James Propp
Subject: Comments on 92.305 assignment #7

- Students would be well-advised to carefully choose the order of logical quantifiers.  "There is $\delta > 0$ such that for all $\epsilon > 0$ is very different from "for all $\epsilon <0$ there exists $\delta > 0$"!

- Some students wrote that a choice for $\delta$ would be choosing it to be greater than a certain quantity.  First of all, if we are to choose $\delta$ according to an inequality, we would want to choose it to be *smaller* than a certain quantity.  For instance, if we find that $\delta = \frac{\epsilon}{|a|}$ is a suitable choice, and we're choosing $\delta$ according to an inequality instead of an inequality, we would need to choose $\delta < \frac{\epsilon}{|a|}$, not $\delta > \frac{\epsilon}{|a|}$.  (Why?)  Second of all, it's general convention to choose $\delta$ to be a *certain* number, instead of one out of many possible choices.  So if we find that $\delta = \frac{\epsilon}{|a|}$ is acceptable, then it's the convention in math to actually choose this as our exact $\delta$.

-Steve