From: Stephen Dalton 
Date: October 16, 2009 5:27:59 PM EDT
To: James Propp 
Subject: Comments on 92.305 assignment #2

- In exercises like 1.3.2, where the text asks you to define something similar to what was defined in the preceding section, students should be careful about copying the solution down from the book.  Many students ended up saying that the infimum of a set $A$ is a number $m$ such that (i) $m$ is a lower bound for $A$, and (ii) if $b$ is any other lower bound for $A$, then $m < b$.  The problem, of course, is in the direction of the inequality at the end; if we want $m$ to be the "greatest lower bound" for $A$ then it will never be less than any other lower bound for $A$!  The problem, I think, was that some students held too fast to Abbott's definition of the *least upper* bound, for which the aforementioned style of inequality would work.

- Performance on exercise 1.4.5 was generally strong.  There was some confusion on exactly how to do it precisely right, but lots of responses specifically stated that it is always possible to choose $n$ large enough so that $1/n < x$ for any $x > 0$.  (In other words, we can make the intervals involved in this problem eventually "dodge" any point we try to say belongs in the intersection by intersecting enough sets.)  Good job!

-Steve