1. Prove the following statements:
(i) every real number is a limit point of Q
(ii) a nite union of closed sets is closed
(iii) if S is a (nonempty) closed and bounded subset of R, then sup S is an element of S.
Solution. (i) Let x R and conside
1. Prove that
Solution. Suppose that 3 = m/n with m, n N having no prime factors in common. Then
3 = m2 /n2 so 3n2 = m2 , which shows m2 is divisible by 3. By Euclids lemma m is also divisible
by 3. So m = 3k for some k N. Now 3n2 = (3k)2
1. A function f : (a, b) R is called uniformly dierentiable if it is dierentiable and for each
> 0, there is > 0 such that x, y (a, b) and 0 < |x y| < imply
f (y) f (x)
f (x) < .
Prove that if f is uniformly dierentiable, then f is uniforml
1. Let f : [a, b] R. Prove the following statements:
(i) If f is continuous and injective, then f is monotone.
(ii) If f is dierentiable and f (x) = 0 for all x (a, b), then f is injective.
(iii) If f is dierentiable and f (a) < 0 < f (b), then
1. Let X be a metric space and suppose Y X. If E Y , we say E is open relative to Y if
for each x E, there is a neighborhood B (x) of x such that B (x) Y E.
Let X = R R, Y = R cfw_0, and E = (0, 1) cfw_0. Give X the usual (absolute value)
1. Dene f : [0, 1] R by f (0) = 0 and
f (x) =
1/n, x = m/n, where m, n N have no common prime factors
Prove that f R and
f dx = 0.
Solution. Fix > 0. Note that S = cfw_x [0, 1] : f (x) /2 is nite; write S = cfw_s1 , . . . , sk .
1. Let f : [a, b] R be a continuous function such that f (a) < 0 < f (b). Prove that there
exists c (a, b) such that f (c) = 0.
Solution. Let S = cfw_x [a, b] : f (x) < 0 and c = sup S. If c = a then f (x) 0 for all x > a.
But then limxa f (x) 0
1. Let cfw_an and cfw_bn be convergent sequences in Rd . Prove that then then lim(an + bn ) =
lim an + lim bn . Prove also that if c R then lim(c an ) = c lim an .
Solution. Assume lim an = a and lim bn = b. Let
> 0. Choose Na , Nb such that
1. The Cantor set consists of real numbers which admit a ternary (base 3) decimal expansion
of the form .a1 a2 a3 . where an cfw_0, 2 for all n. Use this to prove that the Cantor set is
Solution. Let C be the Cantor set and write ea
1. Let cfw_an be a sequence with an 0 for all n. If
an converges, must it be true that
2n a2n also converges? Prove it, or provide a counterexample.
Solution. No. Let an = 1/n if n = 2k for some k = 1, 2, . . ., and otherwise an = 0. Then
Math 5615H Midterm 1
This exam contains 5 questions. Each question is worth 20 points. The exam is worth 100
points in total.
1. Let S be a set, and let T be the set of all functions from S into cfw_0, 1. Prove that 2S = |T |.
1. Let f : S T be a function.
(a) Formulate a condition which is equivalent to injectivity of f , such that your condition
incorporates the image and preimage of f as well as all subsets of S.
(b) Formulate a condition which is equivalent to sur
Math 5615H Midterm 2
This exam contains 4 questions. Each question is worth 25 points. The exam is worth 100
points in total.
1. Let cfw_an be a sequence in a metric space X. Prove that cfw_an is Cauchy if and only if the
Math 5615H Midterm 3
This exam contains 5 pages, with one question per page. Each question is worth 20 points. The
exam is worth 100 points in total.
1. Let f : [0, 1] [0, 1] be continuous. Prove that there is x [0, 1] such that f (x
1. Prove that the Cauchy product of two absolutely convergent series is absolutely convergent.
be absolutely convergent, say
n=0 |an |
= A and
n=0 |bn |
ak bnk .
We must show that cn converges.