Information provided by the grader - thanks!

HW1. I have noticed some of the students are struggling with the definitions
of injective and surjective maps, which caused problems for them in both
section 1.1 and 1.3. They might also benefit from revisiting basic set
theory, specifically showing two sets are equal.

HW2. "Working backwards" to try to solve a problem might
work for some problems, but you cannot write your proof backwards. Start
with what is given/assumed and end with what is trying to be proved.
Recall that "if and only if" proofs require you to do two proofs, where
you swap the hypothesis and conclusion.

HW3. Many people seemed to misunderstand question 2.5.5, and did not
realize they must prove S is the entire real line. Students should review
2.4.2 and 2.4.11 to make sure they understand the rigorous proofs. Note
that infinity is NOT a number and writing "1/infinity" is nonsense.

HW4. Lots of simple mistakes in the limit problems. Double-check
that you answer every part of every problem. The last two problems would
be good to review.

HW5. Dealing with indeterminate forms in limits, eg
"infinity*0", "infinity - infinity", "infinity^0" is problematic. Please
review different approaches you can try for different forms, for example:
factoring, writing as a difference of two squares.
Squeeze theorem is also very useful. Beware of bringing exponents outside
the limit unless they are constants. Problems 9, 14 and 18 are good
practice for this.

HW6. In 7d, a sequence and subsequence share the same limit
only if BOTH converge. In B3, to show a sequence is Cauchy you must find,
for every epsilon >0, a number M such that etc. Choosing epsilon
specifically does not solve the problem, you must use the definition of
Cauchy sequence. In 13, a sequence is r-contractive if there is a fixed
constant 0<r<1 that works for ANY n. You must specify what is r and why it
works for any n.

HW7. Many people did not do or had problems completing C1.
Here is a sketch of the proof, please understand the details. First, show that
(An) and (Bn) converge to the same point, say x. There are many ways to do
this; you may want to prove by induction an equality determining the
length of each nested interval [An, Bn]. Next, prove by induction that y
is contained in the interval [f(An), f(Bn)] for all n. Then y is an upper
bound for lim f(An) and a lower bound for lim f(Bn). Finally use
continuity of f to conclude the proof.

HW8. Almost everyone missed points on #14 where we attempt to prove
uniform continuity on R. You need periodicity for this step, merely being
continuous on R is not enough. To see why, consider f(x)=x^2. It is
continuous, and uniformly continuous on any fixed [a,b], but not on R! For
ANY epsilon and delta you try to pick, you can always find some large x,v
such that |v-x|epsilon. For the problem, use
periodicity to prove you can control the behavior of f(x) on all of R by
"shifting" back to [0,2p] and controlling it there.