C Roettger, Spring ’16
Name (please print): . . . . . . . . .
Math 201 – Practice Questions for Final
Show all work. State clearly what your result is. No cellphones, notes or
books. Time is 120 minutes.
Problem 1 Complete the definition. ”Let S be a set of real numbers. The
real number a is the supremum of S if
a) . . . . . . . . . and
b) . . . . . . . . .
Problem 2 Give an example of a subset S of R which has an infimum but
no minimum.
Problem 3 Consider the set
S=
2 + (−1)n
: n∈N
n
Find the infimum inf S and a sequence in S converging to inf S.
Problem 4 Let (an ) be the sequence defined by a0 = 2 and an+1 = 2an −
n2 + n.
Prove that for all n ∈ N,
an = n 2 + n + 2
Problem 5 A subset A of R is called open if for every x ∈ A, there exists
an open interval around x which is entirely contained in A. Let f : R → R
be a continuous function. Prove that the preimage of every open set if open.
Problem 6 Let A = [0, 1] and B = [0, 1] ∪ {2}. Find a bijection from A to
B.
Problem 7 Define a sequence recursively by setting x0 = 1 and
√ xn
xn+1 = 2 .
Prove that the sequence (xn ) converges to 2. Hint – use induction to prove
that it is increasing and bounded by 2. Then use the Mean Value Theorem
to prove that it indeed converges to 2.