Homework Set #6 Solutions
1. Suppose lim an = +. Show that lim inf an = +.
Proof. Recall if un := infcfw_aj : j n, then un is increasing and lim inf an := lim un .
N such that
Let M > 0 be given. Because lim an = +, there must be an N
Final Exam Solutions
1. Suppose (S, d) is a metric space, and suppose xn is a sequence in S that converges (in
the metric d) to some x S. Show that xn is a Cauchy sequence.
Proof. Let > 0 be given. Since xn x, there is an N such that d(xn , x) <
Homework Set #5 Solutions
1. Suppose an 0, an > 0 for all n N, and suppose we have another sequence sn and
an s R such that |sn s| an for all n N. Show that sn s.
Proof. Let > 0 be given. By assumption, since an 0, there is an N N such that
Homework Set #2 Solutions
1. Suppose that K is an ordered eld.
(a) Implicitly in our several proofs, we used the fact that the zero element (the additive identity) is unique. Prove this. (That is: prove that, if 01 and 02 are both
Homework Set #1 Solutions
2. Suppose we have statements p, q, and r and we know that the following statements
are all true:
(a) p q,
(b) q r,
Does it follow that p is true? Why or why not?
No, it does not follow from these statements th
Homework Set #4 Solutions
1. (a) Show that lim an = a if and only if lim (an a) = 0.
Proof. Suppose an a. Then, since the constant sequence a converges to a, the
addition and subtraction rules for convergent sequences imply that
lim (an a) =
Homework Set #7 Solutions
1. Suppose r 0, r = 1.
(a) Show by induction that
The base case is the statement
rj = r = r
, so the base case is true.
Suppose now that n N is arbi
Mid-Term #1 Solutions
1. Let S R be a non-empty bounded set. Show that there is a sequence an such that
both of the following are true:
(i) an S for all n N.
(ii) lim an = sup S.
For each n N, notice that (sup S)
the set cfw_x S : (sup S)
Homework Set #8 Solutions
1. Give an example of sequence an such that supcfw_aj : j 1 = lim sup an . (Provide
appropriate proof that the supremum and limsup are what you say they are!)
Nice examples are provided by any decreasing convergent seque
Homework Set #3 Solutions
1. Suppose that S and T are non-empty subsets of R, and S T .
(a) Show that inf S inf T .
Proof. We will do this by showing that inf T is a lower bound of S. Let x S
be arbitrary. Therefore, since S T , x T . Since inf T