Practice Problems, Math 220
1 Prove that 52n − 1 is divisible by 8 for all n ∈ N.
2 Prove that
iff a = b.
1
(a
2
+ b)
2
≤
1
2
(a2 + b2 ) for all a, b ∈ R with equality
3 Let
S = {1/n − 1/m : n, m ∈ N} .
Find inf S and supS.
4 Show that a nonempty finite set contains its supremum.
1
5
], Jn = (0, n1 ) and Kn = (n, ∞). Find
n
T∞Let In =T[0,
∞
n=1 Jn and
n=1 Kn .
T∞
n=1 In ,
6 Prove that S 0 is a closed set.
7 If S is a compact subset of R and T is a closed subset of S, prove
that T is compact using (a) the definition of compactness; and (b) the
Heine-Borel Theorem.
1