MATH 3751 Code 2246 Real Analysis: Exam 1 Fall 2002
Name: Score: /100
Please use a pencil and show all work.
1. (10 points) Use induction to prove that
n
X
k=1
k2k= (n−1)2n+1 + 2.
2. (10 points) Prove that for any x, y ∈R,
|x|−|y| ≤ ||x|−|y|| ≤ |x+y|.
3. (10 points) Using the definition of limit, prove that
lim
n→∞
n2
n2+n= 1.
4. (15 points) For each set find its least upper bound and greatest lower bound:
(a) A={1−1/n :n∈N}.
(b) B={1/2,−1/2,3/4,−3/4,7/8,−7/8, . . .}
(c) C={x∈R:x2<2}.
5. (15 points) Let Abe a nonempty bounded set of real numbers.
(a) State the definition of lubA.
(b) Fix k∈R, and set B={x+k:x∈A}.Prove that lubB= lubA+k.
6. (20 points) Answer the following as true or false. If false, give a counterexample.
(a) If {a2
n}converges then {an}is convergent.
(b) If |an| → 0 then an→0.
(c) If α= lubAthen |α|= lub{|x|:x∈A}.
(d) Every convergent sequence of real numbers is bounded.
7. (10 points) Prove that the limit of a convergent sequence is unique.
8. (10 points) Let {xn}and {yn}be sequences. Use the limit laws to prove that if {xn+yn}is
convergent and {xn−yn}is convergent then {xn}is convergent.
