Assignment 4, Math 220
Due: Friday, February 15th, 2002
1 Determine which of the following sets are countably infinite and
which are uncountable:
a: Set of positive rationals
b: Set of all irrationals in (0, 1)
c: Set of all terminating decimals (a decimal whose digits are all
0 from√some point on)
d: {r + n : r ∈ Q and n ∈ N}
e: {x ∈ R−Q : x cannot be written as the square root of a nonnegative rational }
f: {x ∈ R : x is a solution to ax2 + bx + c = 0 for some a, b, c ∈
Q}.
2 Give an example of a countably infinite subset of the set of irrational
numbers which is dense in R.
3 For each sequence below detremine whether it converges and, if it
converges, find its limit. No proofs are required.
a: an = n/(n + 1)
b: bn = (n2 + 3)/(n2 − 3)
c: cn = 2−n
d: tn = 1 + 2/n
e: xn = 73 + (−1)n
f: sn = 21/n
g: yn = n!
h: dn = (−1)n n
i: (−1)n /n
j: (7n3 + 8n)/(2n3 − 31)
k: (9n2 − 18)/(6n + 18)
l: sin(nπ/2)
m: sin(nπ)
n: sin(2nπ/3)
o: (1/n) sin n
p: (2n+1 + 5)/(2n − 7)
q: 3n /n!
r: (1 + 1/n)2
s: (4n2 + 3)/(3n2 − 2)
t: (6n + 4)/(9n2 + 7)
4 Give an example of
1
2
a: a sequence (xn ) of irrational numbers having a limit lim xn that
is a rational number.
b: a sequence (rn ) of rational numbers having a limit lim rn that
is an irrational number.
5 Determine the following limits. No proofs are required, but show
any relevant algebra.
√
a: lim s√
n2 + 1 − n,
n where sn =
2
b: lim(√ n + n − n),
c: lim( 4n2 + n − 2n).